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Basic Geometry : Circles

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Example Question #31 : Circles

The diameter of a circle is 16 centimeters. What is the area of the circle in centimeters?

Possible Answers:

256

50.24

803.84

200.96

Correct answer:

200.96

Explanation:

The area of a circle is found by squaring the circle's radius and multiplying the result by 3.14. The radius is half of the diameter (8 in this case).

$8^{2}\cdot 3.14=64\cdot 3.14=200.96$

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Example Question #73 : Circles

A circle is inscribed in a square whose side is 6 in. What is the difference in area between the square and the circle, rounded to the nearest square inch?

Possible Answers:

$11\ in^{2}$

$8\ in^{2}$

$14\ in^{2}$

$12\ in^{2}$

$4\ in^{2}$

Correct answer:

$8\ in^{2}$

Explanation:

The circle is inscribed in a square when it is drawn within the square so as to touch in as many places as possible. This means that the side of the square is the same as the diameter of the circle.

Let $\pi =3.14$

$A_{square}= s^{2} = (6)^{2} = 36 in^{2}$

$A_{circle}= \pi r^{2}=3.14\cdot3^{2}=3.14\cdot9=28.26 in^{2}$

So the approximate difference is in area $8\ \textup{in}^{2}$

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Example Question #71 : Circles

Two equal circles are cut out of a rectangular sheet of paper with the dimensions 10 by 20. The circles were made to have the greatest possible diameter. What is the approximate area of the paper after the two circles have been cut out?

Figure_2

Possible Answers:

Correct answer:

Explanation:

The length of 20 represents the diameters of both circles. Each circle has a diameter of 10 and since radius is half of the diameter, each circle has a radius of 5. The area of a circle is A = πr² . The area of one circle is 25π. The area of both circles is 50π. The area of the rectangle is (10)(20) = 200. 200 - 50π gives you the area of the paper after the two circles have been cut out. π is about 3.14, so 200 – 50(3.14) = 43.

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Example Question #41 : How To Find The Area Of A Circle

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?

Possible Answers:

325π ft²

525π ft²

175π ft²

275π ft²

125π ft²

Correct answer:

525π 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.

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

$\pi (625-100)$

$525\pi$

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Example Question #21 : Radius

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

Possible Answers:

$4\pi$

$10\pi$

$20\pi$

$25\pi$

Correct answer:

$25\pi$

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

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Example Question #31 : How To Find The Area Of A Circle

What is the ratio of the area of a circle to the area of the square it circumscribes?

Possible Answers:

$\pi \ to\ 2$

$\pi \ to\ 1$

$\pi ^{2}\ to\ 1$

$4\ to\ \pi$

$2\pi \ to\ 1$

Correct answer:

$\pi \ to\ 2$

Explanation:

Since each angle of a square is a right angle, each intercepts a semicircle; as a result, a diagonal of the square doubles as a diameter of the circle. We will call that common measure d=2r , as it will make our ratio easier to calculate.

The area of the circle is $\pi r^{2}$ .

The area of the square is $\frac{1}{2} d^{2} = \frac{1}{2} (2r)^{2} = 2r^{2}$ . This formula is important, even though it is not the more commonly known formula of Area = side².

Therefore the ratio of the area of the circle to that of the square is $\pi r^{2}$ to $2 r^{2}$ , which simplifies to $\pi$ to 2.

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Example Question #31 : Circles

What is the area of a circle with a circumference of 12 inches?

Possible Answers:

$\frac{6}{\pi }\ in^{2}$

$\frac{36}{\pi }\ in^{2}$

$36\pi \ in^{2}$

$\sqrt{6\pi }\ in^{2}$

$6\pi \ in^{2}$

Correct answer:

$\frac{36}{\pi }\ in^{2}$

Explanation:

The relationship of a circumference of a circle to its radius is $C = 2\pi r$ .

Equivalently, $r = \frac{C}{2 \pi }$ .

Set C=12 :

$r = \frac{12}{2 \pi }$

$r = \frac{6}{ \pi }$

To find the area, use the formula $A = \pi r^{2}$ .

Plug in our new value of r:

$A = \pi\left ( \frac{6}{\pi } \right ) ^{2}$

$A = \frac{36}{\pi }$ square inches

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Example Question #83 : Circles

A park wants to build a circular fountain with a walkway around it. The fountain will have a radius of 40 feet, and the walkway is to be 4 feet wide. If the walkway is to be poured at a depth of 1.5 feet, how many cubic feet of concrete must be mixed to make the walkway?

Possible Answers:

$1936\pi \ ft^{3}$

$504\pi \ ft^{3}$

None of the other answers are correct.

$1296\pi \ ft^{3}$

$336\pi \ ft^{3}$

Correct answer:

$504\pi \ ft^{3}$

Explanation:

The following diagram will help to explain the solution:

Foutain

We are searching for the surface area of the shaded region. We can multiply this by the depth (1.5 feet) to find the total volume of this area.

The radius of the outer circle is 44 feet. Therefore its area is 44²π = 1936π. The area of the inner circle is 40²π = 1600π. Therefore the area of the shaded area is 1936π – 1600π = 336π. The volume is 1.5 times this, or 504π.

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Example Question #10 : Area Of A Circle

How many times greater is the area of a circle with a radius of 4in., compared to a circle with a radius of 2in.?

Possible Answers:

$4\pi$

$2$

$2\pi$

$4$

$\pi$

Correct answer:

$4$

Explanation:

The area of a circle can be solved using the equation $A=\pi r^{2}$

The area of a circle with radius 4 is $\pi 4^{2}=16\pi$ while the area of a circle with radius 2 is $\pi 2^{2}=4\pi$ . $16\pi \div 4\pi =4$

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Example Question #81 : Circles

What is the area of a circle whose diameter is 8?

Possible Answers:

16π

8π

12π

32π

64π

Correct answer:

16π

Explanation:

Circarea

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Basic Geometry : Circles

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #31 : Circles

Example Question #73 : Circles

Example Question #71 : Circles

Example Question #41 : How To Find The Area Of A Circle

Example Question #21 : Radius

Example Question #31 : How To Find The Area Of A Circle

Example Question #31 : Circles

Example Question #83 : Circles

Example Question #10 : Area Of A Circle

Example Question #81 : Circles

All Basic Geometry Resources

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