Question

A teelve sided regular polygon is inscribed in a circle of radius 6 cm. what is the area of the twelve sided polygon

Answer 1

Hello there!
_______________________________

We know that the angle subtented by a single side of a twelve sided polygon at its centre is

$\frac{360}{12} = 30$

30 degrees, by the above formula.

Imagine that there are 12 isoceles.triangles, one for each side of the twelve sided polygon subtended at the centre.

Now we find the length of the side of the polygon. Let it be called s. Let r be the radius of the circle, that is 6cm.

$s = 2r \sin( \frac{30}{2} ) \\ s = 2r \sin(15)$

Let the distance of each side from the centre
be h.

$h = r \sin( \frac{30}{2} ) \\ h = r \sin(15)$

Now we find the area of one of the triangles subtended by the sides to the centre, by the formula that the area of the triangle is half the product of it's base and height

$area = \frac{1}{2} \times (s \times h) \\ \\ area = \frac{1}{2} \times (2r \sin(15) \times r \cos(15) ) \\ \\ area = \frac{1}{2} \times {r}^{2} \times 2 \sin(15) \cos(15) \\ \\ area = \frac{1}{2} \times {r}^{2} \times \sin(30) \\ \\ area = \frac{1}{2} \times \frac{1}{2} \times {r}^{2} \\ \\ area = \frac{ {r}^{2} }{4}$

Now we substitute the value of "r" here, which is 6 cm.

$area = \frac{ {6}^{2} }{4} \\ \\ area = \frac{36}{4} \\ \\ area = 9$

Hence we know that the area of the given twelve sided polygon is 9 cm^2
_______________________________

Hope this helps!

Answer 2

In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the reception of such waves and their perception by the brain