Question

Patrick is building a patio in the shape of a decagon with a radius of 12 feet. To determine the amount of cement needed, he must first find the area.

Answer 1

Answer:

The area of the Patrick's building is $2340.22 ft^2$.

Step-by-step explanation:

Shape of the Patrick's building = Decagon

In the figure attached:

Radius of the decagon = r = 12 feet

Side of the decagon = a

Angle AOB = 36°

Using trigonometric ratio is triangle AOC :

AO = r = 12 feet

AC = $\frac{a}{2}=0.5a$

$\tan \theta=\frac{Perpendicular}{base}$

$\tan 36^o=\frac{0.5a}{12 feet}$

$a=\frac{\tan 36^o\times 12 feet}{0.5}=17.44 feet$

Area of decagon :

$A=\frac{5}{2}a^2\times \sqrt{5+2\sqrt{5}}$

$A=\frac{5}{2}(17.44 ft)^2\times \sqrt{5+2\sqrt{5}}$

$A=2340.22 ft^2$

The area of the Patrick's building is $2340.22 ft^2$.