Question

Find the area of the segment of a circle of radius 7 cm whose corresponding sector has a central angle of 60° (π = 3.14).

Answer 1

Given:

Radius r = 7 cm

Central angle θ = 60°

Use π = 3.14

Step-wise calculation:

1. Formula (sector – triangle).

We use A(segment) = `r^2 [ (π xx θ)/360 - (sin θ)/2]`.

2. Substitute r = 7 and θ = 60° (exact form):

A(segment) = `49 xx [ (π xx 60)/360 - (sin 60°)/2]` 

= `49 xx [ π/6 - (sqrt(3)/2)/2]` 

= `49 xx [ π/6 - sqrt(3)/4]` 

= `(49π)/6 - (49sqrt(3))/4` (exact form).

3. Numeric using π = 3.14 and `sqrt(3) ≈ 1.732`:

Sector area = `49 xx π/6` 

= `(49 xx 3.14)/6` 

= `(153.86)/6`

= 25.643333... cm² 

Triangle area = `49 xx sqrt(3)/4` 

= `(49 xx 1.732)/4`

= `(84.868)/4`

= 21.217000... cm² 

Segment area = Sector – Triangle

= 25.643333... – 21.217000... 

= 4.426333... cm²

Exact: A = `(49π)/6 - (49sqrt(3))/4 cm^2`

Numeric (with π = 3.14): A ≈ 4.426 cm²