Teacher’s Guide: Area of an N-gon
Curricular Objectives:
1. Students will learn how incongruent polygons can have similar areas.
2. Students will explore different ways to calculate the area of different polygons.
3. Students will explore ways to change the number of sides of a polygon, but keep
the area of the polygon consistent.
Construct and Investigate:
1. The purpose of this first question is to assess the students’ prior knowledge on
how to calculate the area of a triangle and different quadrilaterals. This first
question also asks students think about how the side lengths of a triangle can be
changed, while the area remains the same.
Triangle
The students’ should remember from previous classes that the area of a triangle
can be calculated be the formula A=(1/2)*b*h, where b is the base of the triangle
and h is the height of the triangle. (as shown)
As long as the length of b and h remain constant, the area of the triangle remains
constant. One way to construct two incongruent triangles with equal area is shown
in the following table.
Construct triangle ABC
Construct a parallel to side
AB through point C
Construct a new triangle
ABC’ by plotting point C’
on the constructed parallel.
Triangle ABC’ maintains a
constant area when point C’
is moved along the
constructed parallel.
Triangles ABC and ABC’
are visually not congruent,
but the area remains the
same because the base and
height of the triangle stays
the same.
2. Kite
3. Students may have trouble with the construction of a kite, if help is needed, a
simple construction is shown below.
Plot two points, A and C
Construct a perpendicular
Plot points B and D on the
bisector to points A and C
perpendicular bisector of
A and C.
Connect consecutive points
ABCD using the polygon
tools (or line segment tool) to
complete the construction.
This construction also
passes the mess up test.
The area of a kite can be measured by dividing the kite into two congruent
triangles with a diagonal line. The area of the kite is twice the area of one of those
triangles (below)
The formula for twice the area of a triangle is A=2*(1/2)b*h=b*h. This means
that the area of a kite can be found by calculating the product of the two
diagonals.
Using the given construction, we can construct a kite that is not congruent but has
the same area by a method similar to our method of construct two triangles that
are not congruent bit have equal area. This construction is shown in the table
below.
Starting with the kite
construction above, we
already have a kite that is
divided into two congruent
triangles.
Construct two parallel lines
to segment BD, one through
point A and one through
point C.
Using the polygon tool, or
segment tool, connect
consecutive points A’BC’D
The two kites have equal
area because the triangles
ABD and A’BD have the
same base and height.
Plot point A’ on one
constructed parallel and
reflect A over the
perpendicular bisector of
BD to point C’.
1. Constructing trapezoids that are not congruent, but have the same area can be
done by finding triangles within a trapezoid.
A general example is shown in the following table.
Start by constructing a
trapezoid using the parallel
line tool.
Connect the two endpoints
of segment AB to the
parallel line to obtain a
trapezoid.
Now, there are infinitely
many triangles that can be
manipulated to change the
congruence of the original
triangle, but not the area.
Here are a few examples.
And a few more
For one specific example
we will use the triangle
ADD’, with point D’ on one
of the two parallel lines.
Consider AD’ to be the base
of the triangle, Find the
point where the extension of
line AB and the parallel of
segment AD’ through point
D intersect. Label to the
intersection point A’.
Now connect consecutive
points A’D’CB to construct
a new trapezoid that is not
congruent to ABCD, but
has the same area.
This is true because triangle
ADD’ and A’AD’ have the
same area and trapezoid
AD’CB remains constant.
Parallelogram
One of the properties of a parallelogram is that each diagonal divides the quadrilateral
into two congruent triangles. Each triangle is a symmetry of the other through the
midpoint of the diagonal. The triangles can be changed, but they have to be
symmetrical. Example below.
Starting with parallelogram
ABCD
Construct a segment
between points A and C
Plot point B’ on the
constructed parallel, to
create triangle AB’C with
area equal to ABC.
To construct our new
parallelogram, perform a
symmetry of triangle AB’C
through the midpoint of any
side of triangle ABC.
Construct parallel to
segment AC through point
B.
Rectangle or Square
We know that our formula for area of a rectangle is A=b*h so as for whatever value
of A we chose, there are infinitely many values for b and h as long as the product is
A. The area for a square is A=b^2, the length of segments can’t be changed. Also,
given the fact that every angle must be right, there are no incongruent squares with
equal area.
Exploration 2
1. Like the previous exploration, we can solve this problem by breaking the decagon
into triangles. Students may immediately see why there is a way to change a
convex decagon into a triangle of equal area, but will have trouble figuring out if
this will work when the decagon is concave.
First look at a quadrilateral. Turning a concave or convex quadrilateral into a
triangle is shown below. The process is the same for a decagon, but it is repeated
many times.
Convex Quadrilateral
Divide into triangle ACD
then find where the parallel
line from the base AC
through point D intersects
the extension of any side.
Use one of the points as for
triangle ACD’ and now
triangle ABD’ has the same
area as ABCD.
Concave Quadrilateral
Divide into triangle ACD
then find where the parallel
line from the base AC
through point D intersects
the extension of any side.
Use one of the points as for
triangle ACD’ and now
triangle ABD’ has the same
area as ABCD.
2. Changing a triangle into a decagon is simply the reversed process of solving for
changing a decagon into a triangle. If students have trouble following the process,
looking back at how to construct incongruent trapezoids with the same area could
help. Because the third point of your triangle will be on one of the triangles sides.
3. The idea here is if a triangle can be changed to have n-many sides and keep the
same area, we can let n=infinity, and the resulting n-gon would be a circle. But,
only if it is a regular n-gon. This process is similar to Rieman Sums and integral
calculus.
Exploration 3
The formula for area of an n-gon is given.
where
S is the length of any side
N is the number of sides
π is PI, approximately 3.142
TAN
http://www.mathopenref.com/polygonregulararea.html
Students are not expected to know or come to this equation. The idea is to have students
understand that the area of a decagon can be broken into the sum of smaller polygons that
are easier to calculate.
2. These measurements are rounded to whole numbers for simplicity reasons. The
idea is to break this into 10 triangles with a base length of 4 and a height of 12.