Question

The area of a circle is 78.5 square centimeters, and a subtending arc on the circle has an arc length of 6. The estimated value of is 3.14. The measure of the angle subtended by the arc is ?

Answer 1

The measure of the angle subtended by the arc is 68.8°

Explanation:

Formula for calculating length of an arc is expressed as:

Length of an arc = theta/360×2πr

Where theta is the angle subtended by the arc

r is the radius of the circle

To get the radius r;

Given Area of the circle to be 78.5cm²

Since area = πr²

78.5 = πr²

78.5 = 3.14r²

r² = 78.5/3.14

r² = 25

r =√25

r = 5cm

This radius of the circle is 5cm

Remember that

Length of an arc = theta/360° × 2πr

6 = theta/360 × 2(3.14)(5)

6 = 31.4theta/360

2160 = 31.4theta

theta = 2160/31.4

theta = 68.8°

Answer 2

The measure of the angle is
`68.79\°`

Step-by-step explanation:

step 1

Find the radius of the circle

The area of the circle is equal to

`A=\pi r^(2)`

we have

`A=78.5\ cm^(2)`

`\pi =3.14`

substitute and solve for r

`78.5=(3.14)r^(2)`

`r^(2)=78.5/(3.14)`

`r=5\ cm`

step 2

Find the circumference of the circle

The circumference of the circle is equal to

`C=2\pi r`

we have

`r=5\ cm`

`\pi =3.14`

Substitute

`C=2(3.14)(5)`

`C=31.4\ cm`

step 3

Find the measure of the angle by an arc length of 6 cm

we know that

The circumference of a circle subtends a central angle of 360 degrees

So

by proportion

`(31.4)/(360)=(6)/(x)\\ \\x=360*6/31.4\\ \\x=68.79\°`