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

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

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 =
`(a)/(2)=0.5a`

`\tan \theta=(Perpendicular)/(base)`

`\tan 36^o=(0.5a)/(12 feet)`

`a=(\tan 36^o* 12 feet)/(0.5)=17.44 feet`

Area of decagon :

`A=(5)/(2)a^2* \sqrt{5+2√(5)}`

`A=(5)/(2)(17.44 ft)^2* \sqrt{5+2√(5)}`

`A=2340.22 ft^2`

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