Find the area of the regular polygon.

Question

Answer 1

A = 3·1/2·18²sin120° = 243√3 ≈ 420..9

Answer 2

The above image is that of a regular triangle, also known as an equilateral triangle.

The distance of 18 is the distance from one point to the centroid.

The line segment shown is 2/3 the total distance from one point to the other flat side.

So the total distance is 3/2 * 18 = 27

This cuts the triangle in half perfectly into two mirror images.

If you separate the triangle into one piece you will see that it is a right triangle.

We know that one side is 27 units.

The hypotenuse is the same distance as all other sides of the triangle.

The remaining side is half the size of the hypotenuse.

Using the Pythagorean theorem and using the variable h for the hypotenuse.

27² + (h/2)² = h²

Simplifying this we get

27² = h² - h²/4

27² = 3h² /4

Taking the square root of both sides yields.

27 = √(3/4) * h

27 = √3 h/2

27*2/√3 = h

*√3 / √3 Eliminating the root in the denominator.

(54*√3) / 3 = h

Now that we know the distance of the hypotenuse and the side length of all sides of the triangle we can solve for the area.

A = base * height / 2

base = h/2

height = 27 (it cuts the triangle it half)

A = 54√3 /3 * 27 / 2

A = 54*27*√3 /6 (use calculator)

A = 243√3

A = 420.9 square units