Example 2

a. Circle? has a radius of 10 cm. What is the area of the circle? Write the formula.

b. What is the area of half of the circle? Write and explain the formula.

c. What is the area of a quarter of the circle? Write and explain the formula.

d. Make a conjecture about how to determine the area of a sector defined by an arc measuring 60°.

e. Circle? has a minor AB with an angle measure of 60°. Sector AOB has an area of 24?. What is the radius of circle

f. Give a general formula for the area of a sector defined by an arc of angle measure x° on a circle of radius?.

Answer: (a) 314.2cm², (b) 157.1cm², (c) 78.55cm² (e) 6.77

Step-by-step explanation: (a) Area of the circle with radius of 10 cm = πr²

= 3.142 * 10 * 10

= 3.142 * 100

= 314.2cm²

The formula = πr²

(b) Area of the half of a circle known as semicircle

= πr²/2

= 3.142 * 10 * 10/2

= 3.142 * 50

= 157.1cm²

The formula = πr²/2

(c) A quarter of a circle is called quadrant

= πr²/4

= 3.142 * 10 * 10/4

= 314.2/4

= 78.55cm²

The formula is written thus = πr²/4, which implies that the circle is divided into 4 unit

(d) The conjecture about how to determine the area of the sector is

Formula of a sector = ∅/360 (πr²)

Information

The arc cant be 60°, therefore information incomplete.

(e) Area of the sector with the angle AOB of 60° = 24.

To find the radius of the angle, make v the subject of the formula from the formula.

Sector area = πr²∅/360°

equate formula to 24.

Therefore πr²∅/360° = 24

Multiply through by360° to make it a linear expression

It now becomes πr²∅ = 24 * 360°

r² = 24 x 360/π * ∅°

r² = 24 * 360° / 3.142 * 60°

r² = 3,640/188.52

r² = 45.8

To find r, we take the square root of both side by applying laws of indicies

Therefore r = √45.8

r = 6.77

(f) General formula = ∅°/360° * (πr²)

angle substended at centre by the arc = x°

assuming the radius of the circle = ycm, Therefore, area of the sector = { ∅°/360° * πy² }