Question

What is the area of a regular hexagon with an apothem of length 16 cm​

Answer 1

Answer:

$A=512\sqrt{3}\ cm^2$

Step-by-step explanation:

we know that

The regular hexagon can be divided into six equilateral triangles

Remember that an equilateral triangle has three equal sides and three equal interior angles (the measure of each interior angle is 60 degrees)

The length side of the triangle is equal to the length side of the regular hexagon

step 1

Find the length side of triangle

Let

b ----> the length side of triangle

see the attached figure to better understand the problem

In the right triangle OAN

Applying the Pythagorean Theorem

$OA^2=AN^2+ON^2$

we have

$OA=b\ cm$

$AN=\frac{b}{2}\ cm$

$ON=16\ cm$ ----> the apothem

substitute

$b^2=(\frac{b}{2})^2+16^2$

$b^2-\frac{b^2}{4}=256$

$\frac{3b^2}{4}=256$

$b=\sqrt{\frac{1,024}{3}}\ cm$

simplify

$b=\frac{32\sqrt{3}}{3}\ cm$

step 2

Find the area of the regular hexagon

Find the area of one triangle and multiply by 6

$A=6[\frac{1}{2}(b)(h)]$

we have'

$b=\frac{32\sqrt{3}}{3}\ cm$

$h=16\ cm$

substitute

$A=6[\frac{1}{2}(\frac{32\sqrt{3}}{3})(16)]$

$A=512\sqrt{3}\ cm^2$