Postulates and Theorems to be Examined in Spherical Geometry.
In forming the foundation on which to build plane geometry, certain terms are accepted
as being undefined, their meanings being intuitively understood. The units that are presented will
accept the following undefined terms:
Point
Line
Lie on
Between
Congruent
Plane
Space.
Terms used in the modules will be defined as follows:
1. Line segment:
The segment AB, AB , consists of the points A and B and
all the points on line AB that are between A and B.
2. Circle:
The set of all points, P, in a plane that are a fixed distance from
a fixed point, O, on that plane, called the center of the circle.
3. Parallel lines:
Two lines, l and m on the plane are parallel if they do not intersect.
4. Sphere:
The locus of the points in space that are a given
distance from a fixed point, called the center of the sphere.
5. Great circle:
A great circle is a circle whose center is the center of
the sphere and whose radius is equal to the radius of the sphere.
6. Arc of a great circle:
The shortest path between two points on the sphere is
the arc of a great circle.
7. Geodesic:
Lines in geometries other than the Euclidean plane. On the sphere
this is a great circle. On the Poincaré model of the hyperbolic plane
it is the inscribed arc of a circle orthogonal to the boundary of the
Poincaré circle.
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8. Antipodal points:
Points that lie at the intersection of a great circle and
a line through the center of the circle on the sphere.
9. Small Triangle:
The small triangle is formed by joining three non-collinear vertices
with the shorter arc between the vertices. Three vertices then
determine only one spherical triangle.
The following postulates will be examined:
1. There exists a unique line through any two points.
2. If A, B, and C are three distinct points lying on the same line, then one and only one of the
points is between the other two.
3. If two lines intersect then their intersection is exactly one point.
4. A line can be extended infinitely.
5. A circle can be drawn with any center and any radius.
6. The Parallel Postulate: If there is a line and a point not on the line, then there is exactly one
line through the point parallel to the given line.
7. The Perpendicular Postulate: If there is a line and a point not on the line, then there is
exactly one line through the point perpendicular to the given line.
8. SAS Congruence Postulate: If two sides and the included angle of one triangle are
congruent respectively to two sides and the included angle of another triangle, then the two
triangles are congruent.
The following theorems will be explored:
1. Vertical Angles Theorem: Vertical angles are congruent.
2. Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two
parallel lines, then it is perpendicular to the other.
3. Theorem: If two lines are parallel to the same line, then they are parallel to each other.
4. Theorem: If two lines are perpendicular to the same line, then they are parallel.
5. Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle is 180o.
6. Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum
of the measures of the two nonadjacent interior angles.
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7. Third Angles Theorem: If two angles of one triangle are congruent to two angles of another
triangle, then the third angles must also be congruent.
8. Angle-Angle Similarity Theorem: If two triangles have their corresponding angles
congruent, then their corresponding sides are in proportion and they are similar.
9. Side-Side-Side (SSS) Congruence Theorem: If three sides of one triangle are congruent to
three sides of a second triangle, then the two triangles are congruent.
10. Angle-Side-Angle (ASA) Congruence Theorem: If two angles and the included side of one
triangle are congruent to two angles and the included side of a second triangle, then the two
triangles are congruent.
11. Theorem of Pythagoras: In a right triangle, the square on the hypotenuse is equal to the sum
of the squares of the legs.
In addition to investigating these postulates and theorems, students will review the fact that
three angles of one triangle congruent to three angles of another is not a condition for congruence
of triangles on the Euclidean plane. They will then investigate whether these conditions lead to
congruence of triangles on the sphere.
As a final activity, students will investigate a formula for the area of a triangle on the
sphere. They will then extend this activity to generate a formula for the area of an n-sided
polygon on the sphere.
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Points, Lines, and Triangles in Spherical Geometry.
This module is intended to be covered over a period of approximately five 45-minute periods.
Objectives:
During this unit, students will
1. Gain an appreciation for the historical background to the development of spherical geometry.
2. Compare their understanding of the terms point, line, and parallel in Euclidean geometry
with what they discover in spherical geometry.
3. Determine which of Euclid’s five postulates are valid in spherical geometry.
4. Determine whether the postulate of betweenness holds in spherical geometry.
5. Construct a spherical ruler.
6. Determine whether vertical angles are congruent on the sphere.
7. Discover that a polygon can be drawn using only two lines on the sphere.
8. Discover that if three segments are drawn from three points that are not collinear to form a
triangle, several possible triangles exist.
9. Establish that the sum of the angles of a triangle on the sphere is more than 1800.
10. Determine whether the measure of the exterior angle of a triangle on the sphere is equal to
the sum of the measures of the two nonadjacent interior angles.
11. Investigate the Third Angles Theorem.
12. Investigate similarity of triangles on the sphere.
13. Investigate congruence of triangles on the sphere.
14. Discover that, unlike the case in Euclidean geometry, AAA is a sufficient condition for
congruence of triangles on the sphere.
15. Derive a formula for the area of the small triangle on the sphere.
For all activities, students may work alone or in pairs. Working in pairs has the advantage
that students can share ideas and ask each other questions as they work through the activities.
It is important that students discover the facts related to spherical geometry using hands-on
manipulatives. Trying to imagine what happens as one works on the sphere has far less value
4
than actually working on one. These manipulatives may be as inexpensive as a ball, some ribbon,
rubber bands, water-soluble markers and pencil and paper. The Lenart Sphere is a more
sophisticated tool that can be purchased through Key Curriculum Press. The sphere comes with a
smooth spherical surface on which students can write with water-soluble pens, a torus on which
to rest the sphere, hemispherical transparencies that fit over the sphere, a spherical
ruler/protractor, and a spherical compass and center locator. While the Lenart sphere offers all
one needs to discover the facts about spherical geometry, one can do just as good a job without
it. Part of the activities presented here offers a suggestion on how to make a simple spherical
ruler/protractor. This manipulative will be quite adequate for the activities suggested here.
Note for the Teacher regarding Historical Background.
More than 2000 years ago the Babylonians, Greeks and Indians were aware of spherical
geometry. In fact, even Euclid made some propositions regarding geometry on the sphere.
Menelaus, a Greek of the first century, published a book Sphaerica, which contains many
theorems about spherical triangles and compares them to triangles on the Euclidean plane. In
these early days, spherical geometry was significant in the study of astronomy and astrology.
Activities:
It would be worthwhile as an introductory activity to give students a globe of the earth
and ask them to find the shortest path for an airplane flying from Washington to Moscow.
Encourage students to place a piece of ribbon on the globe joining the two points and convince
themselves that the shortest path is in fact an arc of a great circle passing through these two
cities. At this stage students should gain an appreciation for the fact that the study of spherical
geometry has direct application in the field of aviation. In addition, spherical geometry has
applications in many other fields of study such as physics, chemistry, and art.
What follows is a series of student-centered activities in which students are actively involved
in discovering similarities and differences between Euclidean geometry and spherical geometry.
The approximate time for each activity is shown in parentheses next to the Note to the Teacher
following each activity. Students should be encouraged to do each activity as it arises and
answer the accompanying questions. The teacher may wish to take time out at the end of each
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period to discuss the students’ observations and get feedback from the students on what they
have discovered.
1. Spherical geometry is geometry on a sphere. What is a sphere?
r
r
Note to the teacher. (5 minutes)
Since students will be working on a sphere it is important that they can define the sphere
in mathematical terms. They should understand that the sphere is the set of all points in
three-dimensional space that are equidistant from a certain fixed point. They should also
understand that in spherical geometry they will be working on the surface of the sphere
and not in the interior of the sphere.
2. In Euclidean geometry we start with a point as one of the undefined terms. Do you think
that we could adopt the point as an undefined term in spherical geometry? Justify your
answer.
6
A
B
Note to the teacher: (5 min)
Point out to students that we need some basic terminology on which to build spherical
geometry just as Euclid had some basic terms on which Euclidean geometry was
developed. We can adopt the point as a basic term in spherical geometry in the same way
as it is adopted as a basic term in Euclidean geometry.
3. a) Locate two points in the plane and label them P and Q. What is the shortest path
between these two points?
Stretch a piece of ribbon between the two points to indicate the shortest path.
P
Q
b) Locate two points on the sphere. (Do not locate these points such that they are
opposite each other on the sphere. Such opposite points are called antipodal points and
they will be referred to later on in the activity.) Label one point A and the other B. Use
a piece of ribbon to find the shortest path between the two points on the sphere just as
you did on the plane. Describe what this path looks like.
7
A
B
Note to the teacher: (5 minutes)
Some students may notice that there are two ways to use their ribbons to find the path
between the two points on the sphere, but one of these presents a shorter path than the
other.
4. a) Extend this segment PQ in the plane. How far can you go?
P
Q
b) Extend the segment on your sphere. How far can you go?
You have just drawn a great circle. Define a great circle.
Name a great circle on Earth.
Are all lines of latitude great circles? Explain your answer.
In the Euclidean plane the shortest path from P to Q is unique, and its measure is fixed.
Can the same be said of the segment AB on the sphere? Is the measure of AB unique?
Explain your answer.
8
A
B
Note to the teacher. (15 minutes)
During the historical background to spherical geometry students became familiar
with the fact that the shortest path ( i.e. the distance) between two cities is not necessarily
a Euclidean line. This activity gives an opportunity to reinforce this idea and students
should be able to determine that the shortest path between two points on the sphere is the
arc of a great circle. Students should be able to define a great circle as the largest circle
on the sphere, or the circle that divides the sphere into two hemispheres. The equator is a
great circle. Lines of Meridian form part of a great circle. Lines of latitude do not form
great circles because they do not divide the earth into hemispheres, but become smaller in
radius as we move away from the equator and towards the poles.
At this stage the term geodesic may be introduced as the line that minimizes the path
between two points. On the sphere this is the great circle.
Students should understand that if P and Q are points on the Euclidean plane, the
shortest path from P to Q (the distance) is unique and is defined as the line segment PQ.
This is not true, however, for points A and B on the sphere. There is not a unique arc of a
geodesic from A to B. In fact, there are at least two geodesic arcs, AB. The distance from
A to B is the length of the shorter arc of the geodesic that passes through A and B.
If A and B are antipodal points then there are an infinite number of geodesics that pass
through A and B.
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5. Measure the distance between two points P and Q in the Euclidean plane. What
instrument did you use to measure the distance?
What units did you measure distance in?
Could you use the same instrument to measure the distance between A and B on the
sphere?
Note to the teacher: (This activity may take up to 45 minutes for students to complete.)
Clearly, it would be difficult for students to get a true measure of the length of the line on the
sphere using a ruler as you did on the Euclidean plane. In spherical geometry the distance
between two points is measured in degrees. Your students will now create their own
spherical ruler that can be used whenever they wish to determine the distance between points
on the sphere.
Firstly we need to understand how distance can be measured in degrees.
The Earth as a sphere in Euclidean space has a radius of 6,400 km i.e. the radius as measured
from the center of the sphere to any point on the surface of Earth is 6,400 km. We may ask
the following questions of the students:
What is Earth’s circumference?
How many degrees does this represent?
If two places on Earth are opposite each other, what is the distance between them in
kilometers in the spherical sense? In degrees?
If two places are 90o apart from each other, how far apart are they in kilometers in the
spherical sense?
If two places are 5026 km apart, what is their distance apart measured in degrees?
Mars has a circumference of 21,321 kilometers. What does this distance represent in
degrees?
What is the furthest distance that two places on Mars can be apart from each other in
degrees? In kilometers (in the spherical sense)?
Now that the students understand why distance on the sphere can be measured in degrees, we
construct a simple spherical ruler for the sphere as follows:
10
Cut strips of paper (the thicker the paper the better since the straight edge of the paper will be
used as a ruler) and lay them side by side so that the strip formed is long enough to form a
great circle. The width of the strips will depend on the size of the sphere you are using. It is
important that the strips be cut thin enough so that when the resulting circle is placed on the
sphere it lies flat and does not have wrinkles in it. The strips can be attached side by side with
pieces of tape.
Once the strips are joined in such a way that they fit snugly around the sphere, we use a
marker to divide the circle into quarters, and label 0o, 90o, 180o, and 270o. By continuing to
fold the circle carefully, it is possible to mark off intervals of 10o on the strip as shown in the
figure below. A semicircle can be attached to the top of the ruler so that it rests firmly on the
sphere while being used.
It is important to note that this ruler can be used as a protractor as well. For this, we mark the
center point at the top of the semicircular section of this ruler. This point is important when
measuring the size of an angle. This point is placed on the vertex of the angle and the
measure of the angle is read off along the circular ruler.
In addition to the ruler/protractor tool that has just been described, a simple protractor can
be made as follows. Cut out a circle on a piece of transparency film. Mark the center of the
circle clearly. Place the circle with its center over a pole on your sphere. Place your ruler
around the equator of the sphere. Tape one end of a piece of string at the pole, and move the
free end of the string around the equator, marking off 100 intervals around the protractor
corresponding with the ten degree intervals around the equator. We now have two useful
tools that can be used to measure the sizes of angles and the lengths of segments on your
sphere.
6. Euclid’s first postulate states that for every point P and every point Q, where P is not
equal to Q, there exists a unique line l through P and Q. Is this postulate valid in
spherical geometry? Justify your answer.
11
Note to the teacher. (10 minutes)
A unique line exists between two points in the Euclidean plane. The same holds true for
points on the sphere except for antipodal points, i.e. those points that lie at the
intersection of a great circle and a line through the center of the sphere. The north and
south poles are examples of antipodal points. Euclid’s first postulate is therefore not valid
in spherical geometry.
7. a) Draw two lines on the plane. In how many points do these two lines
intersect?
b) Draw two great circles on the sphere. In how many points do two lines on the sphere
intersect?
B
A
Note to the teacher. (5 minutes)
Two distinct lines in the plane are either parallel (no common points) or they meet in one
point. On the sphere, any two distinct great circles intersect in two points, so parallel lines
do not exist on the sphere.
12
8. State Euclid’s parallel postulate. How would you re-word this postulate so that it is true
for spherical geometry?
Note to teacher: (5 minutes)
The equivalent of the parallel postulate in spherical geometry states that if l is a line and
P is a point not on l then no line can be drawn through P that is parallel to l.
9. a) Locate a point R between two points P and Q on the plane.
P
R
Q
The Betweenness Axiom states that if P, Q, and R are three points in the plane, then
one and only one point is between the other two.
b) Locate a point C between points A and B on the sphere. Is the Betweenness
Axiom valid for the three points that are drawn on the sphere? Justify your answer.
A
C
B
Note to the teacher: (10 minutes)
Students should discover that while the Betweenness Axiom is valid for Euclidean
geometry, this axiom is not valid on the sphere. If A, B, and C are three points on a great
circle any one of these points can be between the other two.
13
10. Euclid’s second postulate states that a line segment can be extended infinitely from each
side. Is this postulate valid in spherical geometry? Justify your answer.
Note to the teacher. (5 minutes)
The student should note that while Euclidean lines can be extended indefinitely, when the
endpoints of a geodesic are extended, a finite great circle is formed. Euclid’s second
postulate thus fails in spherical geometry.
11. a) Draw a circle on the plane. Locate the center of the circle. Define the term circle.
Measure the radius of the circle.
r
b) Draw a circle on the sphere. Locate the center of the circle. How many centers are
there? Draw in the radius of the circle. Compare the radius of the spherical circle with
that of the planar circle.
Measure the radii of the spherical circle. How is one related to the other?
Note to the teacher. (15 minutes)
Students will discover that while a planar circle has only one center and a radius of
unique length, the same is not true of circles drawn on the sphere. Each circle drawn on
the sphere has two centers. These centers are antipodal points. It is important that
students realize that the radius of the circle drawn on the sphere is itself an arc drawn
along the sphere and not a straight line segment in the Euclidean sense. Every circle that
is not a great circle thus has two radii of different length depending on which center is
being used. If the one radius is equal to r degrees, then the other radius is equal to
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1800 – r degrees. In the case of the circle being a great circle, the two radii are of equal
length, each radius being equal to 90o.
12. Euclid’s third postulate states that a circle can be drawn with any center and any
radius. Is this true for circles on the sphere?
Note to the teacher. (5 minutes)
While students will realize that the above postulate is true on the sphere, the largest circle
that can be drawn on the sphere is a great circle with a radius of 900. There is thus a limit
to the largest circle that can be drawn on the sphere. At this point it may be worth
stressing the difference between what we understand by the term radius of a sphere in the
Euclidean sense, and radius of a circle on the sphere in the spherical sense. In the
Euclidean sense, the radius of the sphere is the distance from the center of the sphere to
any point on the surface of the sphere. However, in the spherical sense, the radius of a
circle drawn on the sphere is the length of the arc drawn from the center of the circle to a
point on the circumference of the spherical circle.
13. a) Draw two intersecting lines on the plane. Use a protractor to measure the vertical
angles. Confirm that the vertical angles theorem is valid on the plane.
P
S
O
R
Q
b) Draw two great circles. Label the points of intersection A and B. Use your spherical
protractor to measure the pairs of vertical formed at the point of intersection of the
great circles. What do you notice?
15
B
A
Note to the teacher. (10 minutes)
Students should note that there are four pairs of vertical angles formed by the intersection
of two great circles. The vertical angles are equal in measure, and the two of the pairs of
vertical angles are congruent.
14. a) Draw a Euclidean line. Locate a point P that is not on the line. What is the shortest
path from the point to the line? This path is called the distance from P to the line.
Construct this path.
How many such paths can you construct?
P
R
S
Q
b) Draw an arc on the sphere. Locate a point P that is not on the arc. What is the
distance (shortest path) from the point to the arc?
How many perpendiculars can be drawn from the point P to the arc?
16
Complete the great circle associated with this arc and locate a pole point for this great
circle. How many perpendiculars can be drawn from this pole to the great circle?
P
Note to the teacher. (10 minutes)
This is a good opportunity for students to review the construction of the perpendicular
from a point to a line. Students should confirm that this shortest path is the perpendicular
drawn from the point to the line.
As in Euclidean geometry, the shortest path from a point on the sphere to an arc is the
perpendicular arc drawn from the point to the arc. Students should note that this
perpendicular can be drawn in at least two ways. (If P is a pole point and the line is an
equatorial line, then there are an infinite number of perpendiculars.) One of these is paths
is shorter than the other. If we were interested in the shortest path, we would measure the
length of the shorter arc from the point to the arc in question.
15. a) Draw two intersecting lines l and m on the plane. Can you draw a common
perpendicular to these two lines?
Draw two parallel lines l and m on the plane. Can you draw a common perpendicular
to these two lines? If you can, how many can be drawn?
l
m
17
b) Draw two great circles that are not perpendicular. Can a perpendicular be drawn to
these two great circles?
Note to the teacher. (10 minutes)
Two intersecting lines in the plane do not have a common perpendicular. Two parallel
lines in the plane have an infinite number of common perpendiculars. On the sphere, two
great circles have a unique common perpendicular. This perpendicular is the great circle
whose pole is at the point of intersection of the other two great circles.
16. a) In how many different ways can three lines intersect in the plane?
l
m
m
k
k
l
b) In how many points do two geodesics intersect? Explain your answer.
In how many ways do three geodesics on the sphere intersect?
18
l
k
m
Note to the teacher. (10 minutes)
Three lines can have zero (parallel lines), one (concurrent lines), two (two parallel lines
and one transversal), or three points of intersection. Three great circles have two points of
intersection if they are concurrent and six points of intersection if they are not concurrent.
17. a) What is the minimum number of sides required to draw a closed figure in the plane
using straight lines only?
Name the figure you drew in the plane.
P
Q
19
R
b) What is the minimum number of sides required to draw a closed figure on the
sphere?
What do you think we should call the figure on the sphere?
You may have decided that the term biangle would be appropriate for this shape. How
many biangles are formed by the intersection of two great circles?
Another name for this figure is a lune. What is the relationship between the two points
of intersection of the sides of the lune?
How long are the sides of the biangle?
Measure the opposite angles of the biangle. What do you notice?
Note to the teacher. (15 minutes)
The minimum number of sides for a Euclidean polygon is three – the triangle. On the
sphere a closed figure can be drawn with only two sides. This figure is called a biangle.
Another name used to describe this shape is the lune. Students should notice that any two
great circles intersect to form a pair of congruent biangles.
The sides of the biangle are semicircles of length 1800 . The opposite angles of the
biangle are congruent and the opposite vertices are antipodal points. We notice that for
any biangle of angle α we may consider another biangle with the same sides and with
angle 360o – α. However, for reasons of consistency, we always consider the small
biangle with angle less than 180o as the biangle under consideration, unless stated
otherwise. We also note that there is not any biangle with angle equal to 180o (since two
great semicircles with angle 180o form a great circle).
20
Students may be interested to know that the word lune is derived from the Latin word
luna which means moon. The lune resembles a waxing or waning moon.
18. a) Construct three non-collinear points in the plane. Connect them to form a triangle.
How many triangles can you form?
b) Locate three non-collinear points A, B, and C, on the sphere. Use different colors to
join two points at a time. How many different arcs join the points together?
How many different triangles with vertices A, B, and C, can be drawn? (Use of
different colors may help to identify the triangles more easily.)
We will define the triangle formed using the shorter arcs joining two points on the
sphere as the ”small triangle.” Identify and shade in the small triangle on the sphere.
A
B
C
Note to the teacher. (15 minutes)
Since two non-antipodal points on the sphere can be joined by two arcs of different
length, a total of 6 arcs of great circles can be identified for the points A, B, and C. These
6 arcs define 8 different triangles on the sphere. The students may wish to use shading to
see these triangles more clearly. It is convenient to choose the shorter arc joining two
points as the side of the triangle. In this case, 3 points on the sphere define a unique
triangle. This enables one to study congruence of triangles on the sphere more easily.
21
19. a) Draw a triangle on the plane. What is the sum of the interior angles of the triangle?
b) Locate three points A, B, and C on the sphere so they form a small triangle.
Measure the angles of the triangle.
What is the sum of the measures of the angles of the triangle?
Draw another larger triangle and measure its angles and find the angle sum.
Is the angle sum the same for both triangles?
Note to the teacher. (20 minutes)
The sum of the angles of a spherical triangle varies from 180o to 5400. Students should
note that the sum of the angles of a triangle on the sphere is not a constant as it is on the
plane.
20. a) Is it possible for a triangle on the plane to have more than one right angle?
b) Is it possible for a triangle on the sphere to have more than one right angle?
22
Note to the teacher. (15 minutes)
A triangle in the plane can have only one right angle.
A triangle on the sphere can have up to three right angles. To draw this triangle, the
student should draw a 900 arc, and from its endpoints, perpendiculars should be drawn to
form the other two sides of the triangle. The angle sum of this triangle is 2700.
21. a) Draw a triangle PQR on the plane. Extend side QR to S. PRS is an exterior angle
of the triangle. What is the relationship between PRS, P, and Q ?
P
Q
S
R
b) Draw ABC on the sphere.
Extend BC to D and measure ACD .
Measure A and B . Is there a relationship between the exterior angle of a triangle
on the sphere and the non-adjacent interior angles?
23
A
B
C
D
Note to the teacher. (15 minutes)
In the plane, the measure of the exterior angle of a triangle is equal to the sum of the
measures of the non-adjacent interior angles. This is not true for triangles on the sphere.
On the sphere, the measure of the exterior angle of the triangle is less than the sum of the
measures of the non-adjacent interior angles.
22. Draw a ΔPQR in the plane. Measure the size of each angle of the triangle. Construct
ΔXYZ with P  X, and Q  Y and XY  2PQ and YZ  2QR . How does the
measure of the third angles of the triangles compare?
How do the measures of PR and XZ compare?
X
P
Z
Q
R
Y
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b) Draw ΔABC on the sphere. Measure the size of each angle of the triangle. Construct
a second ΔDEF with A  D and B  E and DE  2 AB and FE  2BC . Measure
the third angle of the triangle and compare this measure with the measure of the third
angle of triangle ABC.
In the plane, the Third Angles Theorem states that if two angles of one triangle are
congruent to two of another, then the third angles are congruent. Does this theorem
apply to triangles on the sphere?
Note to the teacher. (30 minutes)
The Third Angles Theorem does not apply to triangles drawn on the sphere unless the
two triangles are congruent. Moreover, on the sphere, the larger triangle has a larger
measure for its third angle than the third angle of the smaller triangle.
We also notice that even though the relationship
PQ QR PR


holds in the Euclidean
XY
YZ
XZ
plane, this relationship does not hold on the sphere. Similar triangles thus do not exist on
the sphere unless the triangles are congruent. We will discuss more about this idea later in
the activities.
23. a) State the theorem of Pythagoras with respect to triangle PQR in the plane.
b) Investigate whether this theorem is relevant on the sphere. We have already
discovered that it is possible to draw a triangle on the sphere with one, two, or three
right triangles. Construct one of each of these triangles on the sphere and investigate
whether there is any relationship between the sides of the triangle.
Note to the teacher. (30 minutes)
The student will discover that the theorem of Pythagoras is not relevant on the sphere. In
particular, for the triangle with two right angles at B and C, b = c, and it is not possible to
find a Pythagorean relationship between the sides of the triangle. The lack of relationship
25
is even more evident in a triangle with three right angles since all three sides of the
triangle are congruent.
24. a) What is the definition of similar polygons?
Construct PQR on the plane. Measure its angles and the lengths of the sides.
Construct a second XYZ , with sides double the length of ΔPQR, and with
mP = mX and mQ = mY . Measure the sides of PQR. Find the ratios
PQ QR
PR
,
, and
. What do you notice about the values of the ratios?
XY YZ
XZ
X
P
m YZ = 2.68 inches
m QR = 1.34 inches
m XZ = 2.45 inches
m RP = 1.24 inches
m XY = 2.34 inches
m PQ = 1.17 inches
m XYZ = 58°
m PQR = 58°
m YZX = 54°
m PRQ = 54°
m YXZ = 68°
m QPR = 68°
Q
R
m PQ
= 0.50
m XY
Y
m QR
m YZ
Z
= 0.50
m RP
= 0.51
m XZ
b) Draw a triangle on the sphere. Measure the angles of the triangle and the lengths of
the sides. Construct a second triangle with angles equal in size to the angles of the first
triangle. Measure the sides of the triangle. What do you notice?
You may wish to construct a number of different triangles on the sphere with angles
equal in measure to the angles of the first triangle. In each case measure the lengths of
the sides of the triangles and confirm your initial observations.
26
Note to the teacher.
Two polygons are similar if their angles are congruent and their corresponding sides are
in proportion. Students should confirm that in the plane, if the angles of one triangle are
congruent to the angles of another triangle, then the corresponding sides of the two
triangles are in proportion and the two triangles are similar.
On the sphere, if the angles of one triangle are congruent to the angles of a second
triangle, then their sides are equal in measure. In other words, if two triangles have
congruent angles, then the triangles are congruent. The AAA congruence postulate thus
holds for triangles on the sphere, but does not apply to triangles on the plane. Taken even
further, students should understand that similar triangles on the sphere do not exist.
25. a) Construct a triangle on the plane using convenient lengths for the sides of the
triangle. Construct a second triangle with sides identical in length to the sides of the
first triangle. Are the two triangles congruent?
b) Construct a triangle on the sphere. Measure the length of the sides of the triangle.
Construct a second triangle with sides equal in measure to the sides of the first
triangle. Measure the angles of the two triangles. Are the two triangles congruent?
Note to the teacher.;
SSS is a condition for congruence of triangles on the Euclidean plane. It is also a
condition for congruence of triangles on the sphere.
26. a) Draw ΔPQR on the plane. Measure the length of side PQ and QR and the degree
measure of PQR . Repeat this construction for a second ΔXYZ with the measure
of PQ = XY, QR = YZ, and PQR  XYZ . Are the two triangles congruent?
b) Draw ΔABC on the sphere. Measure the lengths of the sides AB and BC and the
measure of ABC . Repeat this construction for ΔDEF where the measure of AB =
DE, BC = EF and ABC  DEF . Are the two triangles congruent?
27
Note to the teacher:
SAS is a sufficient condition for congruence of triangles on the Euclidean plane and for
small triangles on the sphere.
27. a) Construct ΔPQR on the plane. Measure the length of QR and the measure of
Q and R . Construct ΔXYZ with the measure of QR = YZ and
Q  Y and R  Z . Are the two triangles congruent?
b) Construct ΔABC on the sphere. Measure the length of BC and the measure of
B and C . Construct ΔDEF with the measure of BC = EF and
B  E and C  F . Are the two triangles congruent?
Note to the teacher:
ASA is a sufficient condition for congruence of triangle both on the Euclidean plane and
on the sphere.
28. Write down the formula for the surface area of a sphere.
Construct a biangle with angle of 600. What is the area of this biangle?
Construct a biangle with an angle of 900. What is the area of this biangle?
Write down a generalized formula for the area of a biangle.
60
90
28
Note to the teacher. (30 minutes)
The formula for the surface area of a sphere is A = 4πr2.
For the biangle of angle 60o, the area is
60
1
(4 r 2 ) =
A.
360
6
For the biangle of angle 900 the area is
90
1
(4 r 2 )  A .
360
4
In general , we could say that the area of a biangle of angle  o is

360
(4 r 2 )
Note to the teacher.
The following activity leading to a formula for the area of a triangle on the sphere is a
teacher-led activity. It will take approximately 45 minutes to work through the derivation of
the formula and give the students an opportunity to apply the formula to triangles on the
sphere. Encourage students to participate actively in coloring in lunes on the sphere so that
they can identify the triangles being referenced. Unless students do this, they may not
appreciate the significance of the formula or its derivation.
29. a) Write down the formula for the area of a triangle in the Euclidean plane.
If two triangles have identical angles, are they necessarily congruent?
b) We will now derive a formula for the area of a triangle on the sphere. In order to
understand how the formula is derived, you are encouraged to draw triangles on the
sphere and use colors to identify the different triangles under discussion. This is very
important to a clear understanding of the derivation of the formula for the area of a
triangle on the sphere. This formula is commonly known as Girard’s Theorem.
Draw a triangle on the sphere and label the angles  ,  , and  as shown in th diagram
below.
29
A
a
g
C
b
B
Using colors , draw the α-lune. Notice that there is a congruent α-lune on the back of
the sphere. Repeat this for the β-lune and the γ-lune. Notice that the triangle ABC
appears in each of the lunes. Notice also, that there is a copy of triangle ABC in each of
the lunes on the back of the sphere. If we now wished to get an expression for the area
of the sphere in terms of the area of the lunes, we would get the following (luneα = area
of lune α)
Area of sphere = 2 luneα + 2luneβ + 2luneγ - 4ΔABC
4r 2  2(

360
4ABC  2(
ABC  (
)4r 2  2(

360

180

360
)4r 2  2(
)r 2  (

)4 r 2  2(

360
360
)4 r 2  2(
)r 2  (
180
      180
ABC  r 2 (
)
180


180
)4 r 2  4ABC

360
)4 r 2  4 r 2
)r 2  r 2
This formula has a very interesting consequence for the area of a triangle on the sphere.
It states that the area of a triangle on the sphere is directly related to the angles of the
triangle.
30
The formula
ABC  r 2 (
      180
180
)
is called Girard’s Theorem, and the quantity
      180o
is called the spherical excess of the triangle.
Calculate the area of a spherical triangle with angles of 90o, 90o, and 90p.
Draw these triangles on the sphere and confirm that the answer you got for the area is
consistent with what you would have expected starting with the formula for the sphere,
A  4 r 2
Note to the teacher.
In the Euclidean plane if two triangles have identical angles they do not necessarily have the
same area. These two triangles are similar. On the sphere, however, if two triangles have
identical angles, then they must be congruent. This means that we do not have any noncongruent similar objects on the sphere.
Students should note that the formula for the spherical triangle with angles 90o, 90o, and 90o
is
1
(4  r 2 ) . If they draw the triangle they should note that the triangle formed is in fact one8
eighth of the surface area of the sphere.
A  r 2 (
      180o
)
180o
90o  90o  90o  180o
 r 2 (
)
1800
1
 r 2 ( )
2
1
 (4 r 2 )
8
31
30. Investigate the area of a polygon using the diagram below, and determine whether a
formula for calculating the area of an n-sided polygon exists.
C
1
2
B
1
A
2
E
Note to the teacher.
A general formula for the area of an n-sided polygon may be derived as follows. Firstly,
divide the n-sided polygon into (n-1) triangles as shown in the diagram for the spherical
quadrilateral.
Area of quadrilateral ABCD = Area ABC + Area ACD
o
A1  B  C1  180o
2 A2  C2  D  180
)


r
(
)
180o
180o
A  B  C1  A2  C2  D  360o
 r 2 ( 1
)
180o
=  r2(
But, A1 + A2 = A and C1 + C2 = C , and therefore
Area quadrilateral ABCD =  r 2 (
A  B  C  D  360o
)
180o
A general formula for the area of a spherical polygon with n sides is thus
sum angles of polygon  (n  2)180o
r (
)
180o
2
32
33