# How to Find the Area of a Six-Pointed Star?

## Geometry Math Problem: **Find the area of a six-pointed star which is inscribed inside a regular hexagon**

The figure shows a star that is inscribed inside a regular hexagon. If the sides of the regular hexagon are 10 cm then find the area of the six-pointed star

## Solution

Previously we discussed the area of the five-point star. Here we can solve the problem with the use of symmetry

We can divide the star into 12 equal triangles also all triangles are equilateral triangles

From figure

Area of star = 12 × areas of Triangles

Side of triangle = AC/3

∠ABC = 120°

## To find AC apply cosine rule in ∆ABC

AC² = AB² + BC² – 2×AB×BC×cos 120

⇒ AC² = 100 + 100 – 2×10×10× (-½)

⇒ AC² = 300

Thus, AC = 10√3

So side of triangle = 10√3 /3

⇒ Sides of the triangle = 10/√3

Area of triangle = √3 × (10/√3)²/4

⇒ Area of the triangle = 25/√3 cm²

Then the area of star = 12 × 25/√3

⇒ **Area of star = 100√3 cm²**