Question

select the correct answer.
a regular octagon has side lengths of 8 centimeters. what is the approximate area of the octagon?
a. $512\\,\mathrm{cm}^2$
b. $618\\,\mathrm{cm}^2$
c. $473\\,\mathrm{cm}^2$
d. $309\\,\mathrm{cm}^2$

Explanation:

Step1: Recall area formula for regular octagon

The area $A$ of a regular octagon with side length $s$ is given by:
$$A = 2(1+\sqrt{2})s^2$$

Step2: Substitute $s=8$ cm

Substitute the given side length into the formula:
$$A = 2(1+\sqrt{2})(8)^2$$

Step3: Calculate $(8)^2$ first

$$8^2 = 64$$
$$A = 2(1+\sqrt{2})(64)$$

Step4: Simplify the coefficient

$$2\times64 = 128$$
$$A = 128(1+\sqrt{2})$$

Step5: Approximate $\sqrt{2}\approx1.4142$

$$1+\sqrt{2}\approx1+1.4142=2.4142$$
$$A\approx128\times2.4142$$

Step6: Compute final approximation

$$128\times2.4142\approx309.02$$

Answer:

D. $309\ \text{cm}^2$