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Chapter 5: Problem 51

Verify that for \(n=4\), the formula given by the previous problem reduces to the usual formula for the area of a square.

Short Answer

Expert verified
When we plug in \(n=4\) into the formula for the area of a regular polygon \[\left( A = \frac{n \times s^2}{4\tan(\frac{\pi}{n})} \right)\], we get the expression \[A = \frac{4 \times s^2}{4\tan(\frac{\pi}{4})}\], which simplifies to \(A = s^2\). This matches the usual formula for the area of a square, confirming that the formula does reduce to the formula for the area of a square when \(n=4\).

Step by step solution

01

Find the formula for the area of a polygon

The formula for the area of a regular polygon with side length s and the number of sides n is given by: \[A = \frac{n \times s^2}{4\tan(\frac{\pi}{n})}\]
02

Plug in n=4 into the formula

Now, we will plug n=4 into the formula to see if it matches the area formula for a square: \[A = \frac{4 \times s^2}{4\tan(\frac{\pi}{4})}\]
03

Simplify the expression

We will now simplify the expression: \( \tan(\frac{\pi}{4}) \) is equal to 1. Therefore, \[A = \frac{4 \times s^2}{4 \times 1}\] A = s^2
04

Compare with the formula for the area of a square

The usual formula for the area of a square is given by side^2. As the expression we got above for n=4 is also A = s^2, we can conclude that the formula for the area of a regular polygon does reduce to the usual formula for the area of a square when n=4.

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