Question

Find the area of a regular hexagon with apothem 2√3 mm. Round to the nearest whole number.

Answer 1

Join the center of the hexagon with the 2 base angles.

An equilateral triangle, with side length x, is formed.

(remark: a regular hexagon is made up of 6 equilateral triangles with equal length)

The height $2 \sqrt{3}$ forms 2 congruent right triangles with :

hypotenuse= x, side_1=x/2, and side_2= $2 \sqrt{3}$.

From the pythagorean theorem we have:

$x^{2} = ( \frac{x}{2} )^{2}+(2 \sqrt{3})^{2}$

$x^{2} = \frac{ x^{2} }{4} +12$

$\frac{3}{4} x^{2} =12$

$x^{2} = \frac{12*4}{3}=4*4$

thus, x=4.

The area of the triangle is 1/2 * 4 * $2 \sqrt{3}$=6.93 (mm squared)

The area of the hexagon is 6* the area of the triangle = 42 (mm squared)


Answer: a. 42 (mm squared)