# Area of Regular Six-Pointed Star

**Problem**

Find the area of the regular six-pointed star inscribed in a circle of radius 20 cm.

**Solution**

From the figure shown, angles ADC, AOB, and BOC are equal; all are denoted by θ. See the relationship between inscribed and central angles for detailed explanation about the equality of these angles.

$2\theta = \frac{1}{6}(360^\circ)$

$\theta = 30^\circ$

$\tan \theta = \dfrac{h}{r/2}$

$\tan 30^\circ = \dfrac{h}{20/2}$

$h = 5.7735 \, \text{ cm }$

**Area of triangle ABO:**

$A_{ABO} = \frac{1}{2}rh$

$A_{ABO} = \frac{1}{2}(20)(5.7735)$

$A_{ABO} = 57.735 \, \text{ cm}^2$

**Area of Hexagram (the six-pointed star)**

$A = 12\,A_{ABO}$

$A = 12(57.735)$

$A = 692.82 \, \text{ cm}^2$ *answer*