the number of squares to the number of pentagons is the ratio of what
Answers
Step-by-step explanation:
1
What are the ratios of the areas of equilateral triangles, squares, perfect pentagons, hexagons and heptagons, each with equal sides?
How does the Oppo Reno 5 Pro perform in night photography and videography?
Night or other low light captures have been a pain point for shutterbugs everywhere, but not anymore. Night photography or videogr
Dean Rubine
Answered November 12
Let’s work out the area of a regular n -gon with unit sides.
We’ll call the n vertices A,B,… Since it’s a regular polygon we can circumscribe a circle through the vertices; let’s call its center O.
∠AOB=360∘n
Triangle AOB is isosceles; let’s call M the midpoint of AB; OM bisects ∠AOB is
∠AOM=180∘n
That’s a right triangle, legs AM=12 and OM. We have
tan∠AOM=AMOM
OM=AMtan∠AOM=12tan(180∘/n)
OM is called the apothem; it’s the altitude of triangle AOB. The base is the unit side, length 1, so each triangle has area 14tan(180∘/n), and there are n of those, for a total area of the polygon of
An=n4tan(180∘/n)
Let’s test it out.
Equilateral triangle, A3=34tan60∘=34(3–√)=3–√4✓
Square, A4=44tan45∘=1✓
Pentagon, A5=54tan36∘
In general the ratios go
A3:A4:A5:A6:A7…=3cot60∘:4cot45∘:5cot36∘:6cot30∘:7cot(180∘/7):...:ncot(180∘/n)