In a circle a sector has an area of 16 cm2 and an arc length of 6.0 cm. What is the measure of the central angle in degrees?

Answer 1

To point you in the right direction:

Picture a sector of a circle like a slice of pie. It represents a certain percentage of the whole.

If the interior angle was 90 degrees, it would be exactly one quarter of the pie. 90/360 = 0.25 or 1/4

If the interior angle was 45 degrees, it would be exactly one eighth of the pie. 45/360 = 0.125 or 1/8

So, the interior angle can be described as a fraction of 360 degrees.

To find what that fraction is, we have to use the formulas for the circumference and area of a circle.

A = π*r²

C = 2*π*r

The key is this: the ratio of the interior angle to 360 degrees is equal to the ratio of the area of the sector to the area of the whole circle, which is also equal to ratio of the arc length to the total circumference. So...

θ/360 = 16/A = 6/C

where θ is the interior angle, A is the area of the circle and C is the circumference of the circle. 16 and 6 are given as the area of the sector and the arc length, respectively.

Now if we plug in the formulas for the area and circumference of a circle, we can make solve for r:

6/(2*π*r) = 16/(π*r²)

We have a π*r in both denominators, so they cancel out. simplifying, we get:

3 = 16/r

We can multiply by r and divide by 3 on each side to get r=16/3.

We can then plug r into either the circumference or the area formula and set it equal to θ/360 to solve for the interior angle:

θ/360 = 6/(2*π*r)

θ/360 = 6/(2*π*(16/3))

solving for θ gets us an interior angle of 64.46 degrees.