Question

Circle C is shown. Line segments A C and B C are radii. A line is drawn from point D on line A C to point E on line C B to form a line segment with length 5. The area above the line with length 5 is shaded. The length of D C is 3. Point F is on the outside of the shaded area.

The radius of circle C is 6 units and the measure of central angle ACB is StartFraction pi Over 2 EndFraction radians.


What is the approximate area of the entire circle?
square units

What is the approximate area of the entire sector created by central angle ACB?
square units

What is the approximate area of the shaded region only?
square units

Answer 1

1. The approximate area of the entire circle is 133 square units

2. The approximate area of the entire sector created by the central angle ACB is 28 square units

3. The approximate area of the shaded region only is 22 square units

Answer 2

i. The area of the entire circle is approximately 113 squared units.

ii. Area of the sector ACB is approximately 28 squared units.

iii. Area of the shaded region is approximately 14 squared units.

Explanation:

a. Area of a circle =
`\pi`
`r^(2)`

where r is the radius

The area of the entire circle =
`\pi`
`r^(2)`

=
`\pi` ×
`(6)^(2)`

=
`\pi` × 36

= 36
`\pi` squared units

= 113.143 squared units

The area of the entire circle is approximately 113 squared units.

b. Area of the sector ACB = (θ ÷ 2
`\pi`) ×
`\pi`
`r^(2)`

= (θ ÷ 2
`\pi`) × 36
`\pi`

But, θ =
`(\pi )/(2)` rad. So that:

Area of the sector ACB = (
`(\pi )/(2)`÷ 2
`\pi`) × 36
`\pi`

= 9
`\pi` squared units

= 28.286 squared units

Area of the sector ACB is approximately 28 squared units.

c. Area of the shaded region = Area of triangle CDE

Area of triangle =
`(1)/(2)` × base × height

Draw a perpendicular bisector of DE from C, then apply Pythagoras theorem so that:

h =
`\sqrt{(6)^(2) - (2.5)^(2) }`

=
`√(29.75)`

= 5.453 units

Thus,

Area of the shaded region =
`(1)/(2)` × 5 × 5.453

= 13.633 squared units

Area of the shaded region is approximately 14 squared units.