Comparing Euclidean to Spherical Geometry
Formula
Euclidean
Spherical
Heron’s Formula
A= s( s  a)( s  b)( s  c) where
s = (a+b+c)/2
A = ρ2(A+B+C –π)
where a, b, and c are the lengths of the
sides of a triangle
Pythagorean
Theorem
z2 = x2 + y2
where x and y are lengths of orthogonal
vectors in n-space.
Trig Formulas
O
H
A
cos 
H
O sin 
tan   
A cos
sin  
“SOCATOA” definitions work for
right triangles only!
Law of Sines
Law of Cosines for
Sides
Law of Cosines for
Angles
where A, B, and C are the measurements of
angles of a spherical triangle
cos c = cos a cos b
where a, b, and c are central angle
measurements from respective axes of a sphere
in radians opposite vertex angles A, B, and C,
and C is a right angle
sin a
sin A 
sin c
cos A 
cos a sin b
sin c
tan A 
sin A
cos B
sin A sin B sin C


a
b
c
Definitions work for spherical right triangles
only where a, b, and c are central angle
measurements from respective axes of a sphere
(in radians) opposite vertex angles A, B, and
C; C is a right angle
sin A sin B sin C


sin a sin b sin c
where A, B, and C are angle measures
of any triangle and a, b, and c are
lengths of opposite sides
a2 = b2 + c2 –2bc cosA
where A, B, and C are angle measures of any
spherical triangle, and a, b, and c are angle
measures (in radians) of opposite sides.
cos c = cos a cos b + sin a sin b cos C
where A is an angle measure and a, b,
and c are lengths of sides of any
triangle with side a opposite angle A.
where C is an angle measure of for any
spherical triangle, and a, b, and c are central
angle measures (in radians) of opposite sides.
cos A 
b2  c 2  a 2
2 bc
where A is an angle measure and a, b,
and c are lengths of sides of any
triangle with side a opposite angle A.
cos C = sinA sinB cos c – cos A cos B
where A, B, and C are angle measures for any
spherical triangle, and a, b, and c are central
angle measures (in radians) of opposite sides.
Homework: Fill in the following table. Draw pics to see the triangles.
Warning: The SPT may be hazardous to your health! cos c = cos a cos b
1
2
3
4
5
6
7
8
9
a = π/4
b= π/3
a = π/4
b=
a = π/2
b= π/4
a = π/2
b= π/3
a = π/2
b= π/2
a = π/2
b= 2π/3
Explain what is going on geometrically (picture) in problems 3-6.
triangles satisfy the equation 0 = 0?
Exercise 10.7 p. 214
Exercise 10.8 p. 214
c=
c= π/3
c=
c=
c=
c=
Why do all of these