Math - How to find the area of a circle

A square with a side length of 4 inches is inscribed in a circle, as shown below. What is the area of the unshaded region inside of the circle, in square inches?

Act_math_01

8π - 16

4π-4

8π-4

2π-4

8π-8

Explanation

Using the Pythagorean Theorem, the diameter of the circle (also the diagonal of the square) can be found to be 4√2. Thus, the radius of the circle is half of the diameter, or 2√2. The area of the circle is then π(2√2)2, which equals 8π. Next, the area of the square must be subtracted from the entire circle, yielding an area of 8π-16 square inches.

Circle_graph_area3

100_π_

50_π_

25_π_

10_π_

20_π_

Explanation

Circle_graph_area2

To the nearest tenth, give the area of a circle with diameter $7 \frac{1}{2}$ inches.

$44.2 \textrm{ in}^{2}$

$88.4 \textrm{ in}^{2}$

$22.1 \textrm{ in}^{2}$

$176.7 \textrm{ in}^{2}$

$56.3 \textrm{ in}^{2}$

Explanation

The radius of a circle with diameter $7 \frac{1}{2}$ inches is half that, or $3 \frac{3}{4} = 3.75$ inches. The area of the circle is

$A = \pi r^{2} = \pi \cdot 3.75^{2} \approx 44.2 \textrm{ in}^{2}$

To the nearest tenth, give the area of a circle with diameter 17 inches.

$227.0 \textrm{ in}^{2}$

$907.9\textrm{ in}^{2}$

$289.0 \textrm{ in}^{2}$

$454.0\textrm{ in}^{2}$

$578.0\textrm{ in}^{2}$

Explanation

The radius of a circle with diameter 17 inches is half that, or 8.5 inches. The area of the circle is

$A = \pi r^{2} = \pi \cdot 8.5^{2} \approx 227.0$

To the nearest tenth, give the diameter of a circle with area 100 square inches.

$11.3 \textrm{ in}$

$22.6 \textrm{ in}$

$20.0 \textrm{ in}$

$5.7 \textrm{ in}$

$31.4 \textrm{ in}$

Explanation

The relationship between the radius and the area of a circle can be given as

$A = \pi r^{2}$ .

We can substitute and solve for :

$100= \pi r^{2}$

$r^{2} = \frac{100}{\pi}$

$r=\sqrt{ \frac{100}{\pi}} \approx 5.64$

Double this to get the diameter: $d \approx 5.64 \cdot2 =11.28$ , which we round to 11.3.

Circles

Refer to the above drawing. This shows a ring-shaped garden with inner radius 20 feet and outer radius 40 feet. To the nearest square foot, what is the area of the garden?

$3,770 ; \textrm{ft}^{2}$

$1,257 ; \textrm{ft}^{2}$

$1,885 ; \textrm{ft}^{2}$

$2,514 ; \textrm{ft}^{2}$

$2,827 ; \textrm{ft}^{2}$

Explanation

The total area of the garden is the area of the outer circle - - minus that of the inner circle - .

$A = \pi R^{2} - \pi r^{2} = \pi \cdot 40^{2} - \pi \cdot 20^{2} \approx 5,027 - 1,257 =3,770$

What is the area of a circle with a diameter of ?

$25\pi$

$100\pi$

$5\pi$

$2.5\pi$

$10\pi$

Explanation

The formula for area of a circle is $A=\pi r^2$ . Unfortunately, the problem gives us a diameter instead of a radius. The good news is that the diameter is equal to twice the length of the radius or, mathematically, .

Plug in the given diameter to find the radius:

$\frac{10}{2}=r$

Plug that into our first equation to solve:

$A=\pi r ^2$

$A=\pi 5^2$

$A=25\pi$

Screen_shot_2013-03-18_at_10.29.01_pm

Kate has a ring-shaped lawn which has an inner radius of 10 feet and an outer radius 25 feet. What is the area of her lawn?

125π ft2

175π ft2

525π ft2

275π ft2

325π ft2

Explanation

The area of an annulus is

$\pi (R^{2}-r^{2})$

where is the radius of the larger circle, and is the radius of the smaller circle.

$\pi (25^{2}-10^{2})$

$\pi (625-100)$

$525\pi$

Assume π = 3.14

A man would like to put a circular whirlpool in his backyard. He would like the whirlpool to be six feet wide. His backyard is 8 feet long by 7 feet wide. By state regulation, in order to put a whirlpool in a backyard space, the space must be 1.5 times bigger than the pool. Can the man legally install the whirlpool?

Yes, because the area of the whirlpool is 18.84 square feet and 1.5 times its area would be less than the area of the backyard.

Yes, because the area of the whirlpool is 28.26 square feet and 1.5 times its area would be less than the area of the backyard.

No, because the area of the backyard is smaller than the area of the whirlpool.

No, because the area of the whirlpool is 42.39 square feet and 1.5 times its area would be greater than the area of the backyard.

No, because the area of the backyard is 30 square feet and therefore the whirlpool is too big to meet the legal requirement.

Explanation

If you answered that the whirlpool’s area is 18.84 feet and therefore fits, you are incorrect because 18.84 is the circumference of the whirlpool, not the area.

If you answered that the area of the whirlpool is 56.52 feet, you multiplied the area of the whirlpool by 1.5 and assumed that that was the correct area, not the legal limit.

If you answered that the area of the backyard was smaller than the area of the whirlpool, you did not calculate area correctly.

And if you thought the area of the backyard was 30 feet, you found the perimeter of the backyard, not the area.

The correct answer is that the area of the whirlpool is 28.26 feet and, when multiplied by 1.5 = 42.39, which is smaller than the area of the backyard, which is 56 square feet.

A 6 by 8 rectangle is inscribed in a circle. What is the area of the circle?

$20\pi$

$10\pi$

$4\pi$

$25\pi$

Explanation

The image below shows the rectangle inscribed in the circle. Dividing the rectangle into two triangles allows us to find the diameter of the circle, which is equal to the length of the line we drew. Using a2+b2= c2 we get 62 + 82 = c2. c2 = 100, so c = 10. The area of a circle is $A=\pi r^2$ . Radius is half of the diameter of the circle (which we know is 10), so r = 5.

$A=\pi *5^2=25\pi$

Diagram_1

How to find the area of a circle

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