Day 6 Problem

Holovision needs to design a 50-sided regular polygon for his new holographic project. Given any four consecutive vertices EFGH, the distance of E from H is 36 centimeters. What is the area of the polygon?

The problem was originally published at @savvyplayer/mayday6math.

Answer

28,914 cm² 🎯

Due to possible rounding off errors, answers very close to that value will be considered correct.

Solution

There are multiple possible solutions to this problem which will give the same answer. This solution is just one of them.

These are the given facts on the original problem:

• The polygon has 50 equal sides.
• The distance between E and H is 36 centimeters.

From those facts, we can obtain the following:

• An interior angle of the polygon measures 172.8°. Angles EFG and FGH have that measure.
• Angles FEG and GHE measure 7.2°, which is the supplementary angle of 172.8°.

From that point, here is the solution:

• Let variable s be the measure of any side of the polygon.
• Assume there is a line segment GX perpendicular to EH, forming a triangle GXH were X is the right angle and GH = s. Side HX measures one-half of the difference of the measures of EH and s (you may refer to the image), which is 18 cm - s/2.
• Since we have the measures of GH and XH relative to s, we can use the Pythagorean theorem to get the measure of GH, also relative to s, which results to √(3s²/4 - 324). (valid but unnecessary)
• To get the value of s, we can use the trigonometric ratio cos (x) = (adjacent side) / (hypotenuse), which in our problem is cos (7.2°) = (18 - s/2) / s, which when solved for s gives s ≈ 12.0634 cm.

The formula for the area of a regular polygon given the number of sides and the length of a side, which is the following:

A = n * s² * cot (π/n) / 4

where

• A = area of the regular polygon
• n = number of sides
• s = length of a side

Since we already have the number of sides of the polygon and its side length, we can use the formula above to get the area of the polygon. Note that since there was rounding off in this solution before the final answer was obtained, there are noticeable tiny errors in rounding off.

• A ≈ 50 * 12.0634² * (cot (π/50) / 4
• A ≈ 50 * 145.5256 * 15.8945 / 4
• A ≈ 28913.2897

The more accurate value of the area of the polygon when the equation is directly entered onto a scientific calculator is 28,913.94 (square centimeters). 🤓

Winner: @minus-pi 🏅

1 HIVE has been sent to @minus-pi's Hive account. 💰

This problem was indeed a tough one, but @minus-pi still managed to get the correct answer! 😅

 
Mentions: @jfang003, @holovision, @eturnerx (@eturnerx-dbuzz), and @ahmadmanga (@ahmadmangazap) 🤓
Special mentions: @dbuzz, @chrisrice, @jancharlest, and @mehmetfix 🤯