# a central angle in a circle has a measure of 34.4 and the measure of the area of its sector is 139.6 square units. what is the radius?

full question:

a central angle in a circle has a measure of 34.4 and the measure of the area of its sector is 139.6 square units. to the nearest tenth of a unit, what is the measure of the circle's radius?a. 19.4b. 21.6c. 27.3d. 29.7

Relevance

The area of a full circle is:

A = πr²

Given a full circle is 360°, the area of a sector (with central angle θ) is a fraction of full circle --> θ/360°

So the area of a sector is:

A(sector) = θ/360° * πr²

139.6 = 34.4/360 * πr²

r² = 139.6 * 360/(34.4π)

r² ≈ 465.03

r ≈ √465.03

r ≈ 21.6 units

• 34.4 without dimensions should be interpreted as radians except I'm fairly sure you meant 34.4°.

Working in radians, the relationship between central angle, Θ, radius, r, and circular sector area, A, can be stated as:A = Θr²/2

Converting the angle to degrees, use the conversion:

or

Using this conversion, and working in degrees:

A = (π/180)Θr²/2 = (Θ/360)πr²

In either case, you should be able to recognize that the sector area is the same proportion of the area of a full circle as the sector angle is to the angular dimension of a full circle.

As/Ac = Θ/(2π) = Θ°/360°

Plugging in the given values:

A = (34.4/360)πr² = 139.6

r = √[139.6*360/(34.4π)]

r = 21.6 (rounded to 3 sig. dig)

• Sector area A = (θ/360)πr^2

r^2 = (360/θ)(A/π) = (360/34.4)(139.6/π) ~ 465.02

r ~ 21.56 units, (nearest to b)

• Area of sector working in degrees

A = (x/360) pi r^2

x = central angle

Plug in A and x and solve for r