Question

A regular 24-gon has a side length of 4 km. Find the measure of one-interior angle, the perimeter and the area of the 24-gon

Answer 1

α = 165°

P = 96 km

A = 96√3 km^2

Explanation:

Given:

A regular 24-gon

a = 4 km (side length)

n = 24 (the number of angles and sides)

Find: α - ? (interior angle)

P - ? (perimeter)

A - ? (area)

First, we can find the sum of the angles of a 24-gon:

`s = (n - 2) * 180°`

`s = (24 - 2) * 180°`

`s = 22 * 180 = 3960°`

`\alpha = (3960°)/(24) = 165°`

`p = 4 * 24 = 96 \: km`

`a = n * \frac{ {a}^(2) * √( 3) }{4}`

`a = 24 * \frac{ {4}^(2) * √(3) }{4} = 6 * 16 √(3) = 96 √(3) \: {km}^(2)`

Answer 2

well, let's move like the crab, let's get the perimeter first, hmmm well, we know each side is 4 km, and is a 24-gon, so it has 24 sides, so the perimeter is just the sum of all those 24 sides, or we can say (4)(24) = 96 km.

now, let's get the interior angle of it

`\underset{in~degrees}{\textit{sum of all interior angles}}\\\\ n\theta = 180(n-2) ~~ \begin{cases} n=\stackrel{number~of}{sides}\\ \theta = \stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ n=24 \end{cases}\implies 24\theta =180(24-2) \\\\\\ 24\theta =180(22)\implies \theta =\cfrac{180(22)}{24}\implies \theta = 165`