Question

A circle with radius of \greenD{2\,\text{cm}}2cmstart color #1fab54, 2, start text, c, m, end text, end color #1fab54 sits inside a \blueD{7\,\text{cm} \times 11\,\text{cm}}7cm×11cmstart color #11accd, 7, start text, c, m, end text, times, 11, start text, c, m, end text, end color #11accd rectangle. What is the area of the shaded region? Round your final answer to the nearest hundredth.

Answer 1

64.43cm ^2

Explanation:

First, calculate the area of the whole figure, including the unshaded area.

The area of a rectangle is the length times the width.

7cm x 11cm = 77c
`m^(2)`

Next, calculate the area of the inner figure.

The area of a circle is
`\pi`
`r^(2)`

`\pi` x 2cm x 2cm = 4
`\pi` c
`m^(2)`

Finally, subtract the area of the inner circle from the area of the outer rectangle.

77 c
`m^(2)` - 4
`\pi` c
`m^(2)` ≈ 64.43 c
`m^(2)`

Answer 2

• 64.43 cm^2
• 64.44 cm^2 using π = 3.14

Explanation:

The area of the 7 cm by 11 cm rectangle is ...

A = bh

A = (7 cm)(11 cm) = 77 cm^2

The area of the circle of radius 2 cm is ...

A = πr^2 = π(2 cm)^2 = 4π cm^2

If the shaded area lies between the circle and the rectangle, its area is the difference of these:

shaded area = 77 cm^2 -4π cm^2

Using π = 3.14, this area is ...

(77 -4·3.14) cm^2 = 64.44 cm^2

Using π = 3.141592, this area is ...

(77 -4·3.141592) cm^2 ≈ 64.43 cm^2

The shaded area is 64.43 cm^2, (64.44 if you use π=3.14).

_____

Comment on π

Often, you are required to use a specific valued for π, even if that is inappropriate for the number of significant digits required in the answer. In recognition of that, we have offered both the correct answer (64.43) and the one associated with the value of π you may be expected to use.