Basic Geometry : How to find the area of a circle

Study concepts, example questions & explanations for Basic Geometry

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Example Questions

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Example Question #1 : Plane Geometry

Screen_shot_2013-09-16_at_1.04.39_pm

The radius of a circle is 4 cm, what is the area?

Possible Answers:

Correct answer:

Explanation:

The area of a circle is found by: , where r is the radius.

.

The area of the circle is .

Example Question #1 : Circles

The radius, , of the circle below is 18 units. What is the area of the circle?

Circle

Possible Answers:

 square units

Cannot be determined

 square units

 square units

 square units

Correct answer:

 square units

Explanation:

The formula for the area, , of a circle with radius  is:

We can fill in 

You could do the arithmetic to get an area of about 1,017.876 square units, but it is ok and more precise to leave it as shown.

Example Question #1 : How To Find The Area Of A Circle

Give the area of a circle with diameter 13.

Possible Answers:

Correct answer:

Explanation:

Half of the diameter 13 is the radius . Use the area formula:

Example Question #2 : How To Find The Area Of A Circle

Figure2

Point A is the center of the circle above.

Figure ABCD is a square.

Segments AB and AD are radii of the circle.

The radius of the circle is  units.

Find the area of the red-colored shape.

Possible Answers:

Correct answer:

Explanation:

Since it is stated that A is the center of the circle and ABCD is a square, angle BAD is a right angle that comprises exactly one fourth of the circle (90o is one fourth of 360o). This can also be seen through a proportion.

Since the angle that contains the red area is one fourth of the circle, finding the area of the entire circle, and then one fourth of that number gives the answer. Use the equation  to find the area of a circle, where  is area and  is the radius.

Example Question #5 : Plane Geometry

Figure4

Find the area of the ice rink formed by the rectangle and semi-circles with the dimensions shown.

Possible Answers:

Correct answer:

Explanation:

To find the area of the entire rink, find the area of the rectangle in the middle of the rink and add it to the area of the circle that would be formed by the two semi-crcles on the edges of that rectangle.

The area of the rectangle is found by multiplying its width by its length.

To find the area of the circle that accounts for the two semi-circles, use the formula , where  is the radius of the circle.

The radius of the circle is half of its diameter. The diameter of the circle is the same as the width of the rectangle in the case of this rink, and half of that is .

Adding this area to the area of the rectangle gives the area of the entire rink.

Example Question #1 : How To Find The Area Of A Circle

If a circle has circumference 8π, what is its area?

 

Possible Answers:

4

16π

8

Correct answer:

16π

Explanation:

If the circumference is 8π, then since C=2πr we know r=4. We further know that A=πr2, so A=π(4)2=16π.

 

Example Question #1 : How To Find The Area Of A Circle

If the equation of a circle is (x – 7)+ (y + 1)= 81, what is the area of the circle?

Possible Answers:

49π

6561π

81π

18π

Correct answer:

81π

Explanation:

The equation is already in a circle equation, and the right side of the equation stands for r2 → r= 81 and r = 9

The area of a circle is πr2, so the area of this circle is 81π.

Example Question #6 : How To Find The Area Of A Figure

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? 

Possible Answers:

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.

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.

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.

Correct answer:

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.

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. 

Example Question #1 : How To Find The Area Of A Circle

There are two identical circles on a plane that overlap. The radius of both circles is 1. The region in which they overlap has an area of π.

What is the total area of the 2 overlapping circles?

Possible Answers:

π

0

1

2π

2

Correct answer:

π

Explanation:

The total area of both circles is π + π = 2π

Since the region overlaps, we cannot count it twice, so we must subtract it.

we get 2π – π = π

Example Question #5 : How To Find The Area Of A Figure

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

Possible Answers:

2π-4

8π-4

8π - 16

8π-8

4π-4

Correct answer:

8π - 16

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.

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