Question

Find the area of a regular pentagon with apothem length of about 10.3 meters. Round to the nearest tenth if necessary.

Options:

77.3 units2

386.3 units2

154.5 units2

772.5 units2

Answer 1

The answer is 386.3 units ^2

Step-by-step explanation: i just took the test and got it correct!

Answer 2

The area = 182.52 sq units.

Explanation:

1. Area of regular polygon can be found by splitting into smaller triangles.
2. Each triangle has base as side of polygon and vertex as center of polygon.

now consider ΔOAB,

we know ∠AOB =
`(360)/(5)`° =72°

(this is due to symmetry)

as ∠AOD = (∠AOB)/2 = 36°

Now we know AD=AB/2 =
`(10.3)/(2)` =5.15 m

using trignometric relation tan(∠AOD) =
`(AD)/(OD)`

⇒OD =
`(AD)/(tan(36))` =
`(5.15)/(tan(36))`

AREA OF TRIANGLE = BASE × HEIGHT /2

area of ΔOAD =
`(1)/(2)` × 5.15 ×
`(5.15)/(tan(36))`

=
`((5.15)^2)/(2tan(36))`

As there are 10 such triangles like ΔOAD

Total area =10×
`((5.15)^2)/(2tan(36))`

= 182.525447428574