Suggested problems - solutions
Polygons in Euclidean and hyperbolic geometry
P1: Which of the hyperbolic polygons are convex?
The quadrilateral (4-gon) GHIJ is convex, the 5-gon KLMNO is convex, the 6-gon ABCDE
is not - the interior of angle CDE faces outward (i.e., the polygon is not contained by all its
interior angles).
P2: In Euclidean geometry, compute
(a) The angle sum of a convex hexagon (6-gon).
n = 6, so angle sum = (6 − 2)(180)◦ = 720◦. (This formula does not require a regular
hexagon; you can always get the sum... but if it isn’t regular, there’s no way to figure out
what the individual angles measure.)
(b) The measure of an interior angle if the hexagon is regular.
If regular, you can divide that sum by six: each (interior) (vertex) angle measures 120◦
(c) The measure of a central angle (if regular)
Central angles always sum to 360◦ , and this holds in both Euclidean and hyperbolic - it’s
based off of angle measure / linear pairs supplementary and so on.
Each central angle measures 60◦ . (360◦ divided by six angles)
(d) If the side length of a regular hexagon measures 7, what is the length of the apothem?
Start with θ = half a central angle: θ = 30◦ . Then
a = .5(s) cot θ =
.5(7)
.5(s)
=
≈ 6.06
tan θ
tan 30◦
(e) What is the value of the perimeter?
Perimeter is sum of sides (distance around edge). P = 6(7) = 42.
(f) What is the value of the area?
A = .5(a)(s)(n) = .5(6.06)(7)(6) = 127.26
P3: If you want more practice, do the above for a decagon (10-gon).
(a) The angle sum of a convex decagon (10-gon).
n = 10, so angle sum = (10 − 2)(180)◦ = 1440◦ .
(b) The measure of an interior angle if the decagon is regular.
If regular, you can divide that sum by ten: each (interior) (vertex) angle measures 144◦
(c) The measure of a central angle (if regular)
Each central angle measures 36◦ . (360◦ divided by ten angles)
(d) If the side length of a regular decagon measures 7, what is the length of the apothem?
Start with θ = half a central angle: θ = 18◦ . Then
a = .5(s) cot θ =
.5(s)
.5(7)
≈ 10.77
=
tan θ
tan 18◦
(e) What is the value of the perimeter?
P = 10(7) = 70.
(f) What is the value of the area?
A = .5(a)(s)(n) = .5(10.77)(7)(10) = 376.95
P4: Compute the value of the defect, the area, and the perimeter for the hyperbolic 5-gon shown
below.
Find the sum of the angle sums of the triangles (which gives the angle sum for the 5-gon):
(101.6 + 80.8 + 101)◦ = 283.4◦
Find the angle sum for a Euclidean 5-gon: (5 − 2)(180) = 540◦ .
The defect is the difference: δ = (540 − 283.4)◦ = 256.6◦ .
π (256.6) ≈ 4.49.
The area is A = 180
And the perimeter is the distance around the outside edge - there’s no formula for it, other than
“add the edges”
AB ∗ + BC ∗ + CD∗ + DE ∗ + EA∗ = 2.7 + 1.855 + 2.433 + 2.418 + 2.048 = 11.454
Non-Euclid is calculating these distances in the hyperbolic metric - it would be very very tedious
if everything had to be worked out by hand - and you’d need coordinate points. The only
reasonable perimeter problem is one where the side lengths are simply given.