Question

Find the area of the shaded regions. Give your answer as a completely simplified exact value in terms of π (no approximations).

Answer 1

Answer: (40/3)pi

This is the same as (40pi)/3

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Explanation:

A = area of the smaller circle OB

A = pi*r^2

A = pi*3^2

A = 9pi

Circle OB has an exact area of 9pi square cm

We don't want the whole circle area, since we only are focusing on a 120 degree slice. Note how 120 degrees is 120/360 = 1/3 of a full circle; therefore we want 1/3 of the area to get the area of the slice

area of slice DOB = (1/3)*(area of circle OB)

area of slice DOB = (1/3)(9pi)

area of slice DOB = 3pi

We'll use this later so let's call this M

M = 3pi

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B = area of larger circle OA

B = pi*r^2

B = pi*7^2

B = 49pi

We take 1/3 of this for similar reasons as mentioned earlier. So the area of pizza slice COA is (1/3)*(49pi) = 49pi/3

We'll use this later so let N = 49pi/3

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Subtract the values of N and M.

Shaded area = N - M

Shaded area = (49pi/3) - (3pi)

Shaded area = (49pi/3) - (9pi/3)

Shaded area = (49/3)pi - (9/3)pi

Shaded area = (49/3 - 9/3)pi

Shaded area = (40/3)pi

side note: I think you computed 49/3 = 16.333 repeating which was the result for the area of circle OA; however it's not the final answer

another note: leave 40/3 as a fraction and don't convert it to 13.333 repeating. This is because the fraction 40/3 is exact while 13.333 is approximate