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Basic Geometry : Circles

Study concepts, example questions & explanations for Basic Geometry

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9 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #21 : How To Find The Length Of A Radius

Find the length of the radius of a circle inscribed in a square that has a diagonal of $10\sqrt2$ .

Possible Answers:

$5\sqrt2$

Correct answer:

Explanation:

Notice that the diagonal of the square is also the hypotenuse of a right isosceles triangle whose legs are also the sides of the square. You should also notice that the diameter of the circle has the same length as that of a side of the square.

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Now, substitute in the value of the diagonal to find the length of a side of the square.

$\text{side}=\frac{10\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=10$

Now keep in mind the following relationship between the diameter and the side of the square:

$\text{Diameter}=\text{side}=10$

Recall the relationship between the diameter and the radius.

$\text{Radius}=\frac{1}{2}(\text{Diameter})$

Substitute in the value of the radius by plugging in the value of the diameter.

$\text{Radius}=\frac{1}{2}(10)$

Solve.

$\text{Radius}=5$

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Example Question #23 : How To Find The Length Of A Radius

Find the length of the radius of a circle inscribed in a square that has a diagonal of $14\sqrt2$ .

Possible Answers:

$28\sqrt2$

$\frac{7}{2}$

Correct answer:

Explanation:

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Now, substitute in the value of the diagonal to find the length of a side of the square.

$\text{side}=\frac{14\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=14$

Now keep in mind the following relationship between the diameter and the side of the square:

$\text{Diameter}=\text{side}=14$

Recall the relationship between the diameter and the radius.

$\text{Radius}=\frac{1}{2}(\text{Diameter})$

Substitute in the value of the radius by plugging in the value of the diameter.

$\text{Radius}=\frac{1}{2}(14)$

Solve.

$\text{Radius}=7$

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Example Question #24 : How To Find The Length Of A Radius

Find the length of the radius of a circle inscribed in a square that has a diagonal of $50\sqrt2$ .

Possible Answers:

$50\sqrt2$

$25\sqrt2$

Correct answer:

Explanation:

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Now, substitute in the value of the diagonal to find the length of a side of the square.

$\text{side}=\frac{50\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=50$

Now keep in mind the following relationship between the diameter and the side of the square:

$\text{Diameter}=\text{side}=50$

Recall the relationship between the diameter and the radius.

$\text{Radius}=\frac{1}{2}(\text{Diameter})$

Substitute in the value of the radius by plugging in the value of the diameter.

$\text{Radius}=\frac{1}{2}(50)$

Solve.

$\text{Radius}=25$

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Example Question #22 : How To Find The Length Of A Radius

Find the radius of a circle given the circumference is $8\pi$ .

Possible Answers:

$4\pi$

$16\pi$

$8\pi$

Correct answer:

Explanation:

To solve, simply use the formula for the circumference of a circle and solve for r. Thus,

$C=2\pi{r}\Rightarrow r=\frac{C}{2\pi}$

$r=\frac{8\pi}{2\pi}=4$

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Example Question #23 : How To Find The Length Of A Radius

Find the length of the radius of a circle given the diameter is 3.

Possible Answers:

$\frac{3}{2}$

unknown

$9\pi$

Correct answer:

$\frac{3}{2}$

Explanation:

To solve, simply use the formula for the diameter and solve for r. Thus,

d=2r

Therefore, when solved for r,

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

Plug in d and:

$r=\frac{3}{2}$

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Example Question #24 : How To Find The Length Of A Radius

A circle has a circumference of $32 \pi$ inches. What is the radius of the circle?

Possible Answers:

$32\textup{ in}$

$4\textup{ in}$

$16\textup{ in}$

$4\sqrt2\textup{ in}$

$16\pi\textup{ in}$

Correct answer:

$16\textup{ in}$

Explanation:

The circumference of a circle is given by $c = 2\pi * r$ , where is the circumference and is the radius.

Plug in the given circumference for and solve for :

$32 \pi = 2\pi * r$

$\frac{(32 \pi)}{2\pi} = \frac{(2\pi * r)}{2\pi}$

$r= 16\ in$

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Example Question #184 : Basic Geometry

Oscar's bike will travel {70}'' for the wheels on his bike to make one complete rotation. What is the radius of the wheels on Oscar's bike? Round to the nearest inch?

Possible Answers:

{22}''

{13}''

{18}''

{9}''

{11}''

Correct answer:

{11}''

Explanation:

The distance that the bicycle travels for one full rotation of the wheels represents the circumference of the wheel. Therefore, if {70}'' is travelled for the wheel to rotate once completely, then the circumference for the bike wheel is {70}'' . With this information, we can use our circumference formula to determine the radius of the wheel:

$\\circ.=\pi\cdot diameter\\{70}''=3.14159\cdot diameter\\diameter=\frac{{70}''}{3.14159}={22.28}''\approx{22}''$

Since radius is half the diameter, we can conclude that the radius of the wheel can be rounded to {11}'' .

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Example Question #24 : How To Find The Length Of A Radius

What is the radius of a circle whose area is $36\pi$ ?

Possible Answers:

Correct answer:

Explanation:

The formula to calculate area is,

$A=\pi r^2$ .

We are told that the area is $36\pi$ so we set the

$36\pi=\pi r^2$

diving each side by $\pi$ gives us

36=r^2

take the square root of both sides and we see that

r=6

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Example Question #30 : How To Find The Length Of A Radius

Michael's Clock has an area of $169\pi\ in^2$ . Find the radius.

Possible Answers:

$15\ in$

$12.5\ in$

$17\ in$

$13\ in$

$11\ in$

Correct answer:

$13\ in$

Explanation:

The equation to find any circle is

$\pi r^2$

The total area is 169 inches^2

$\pi r^2 = 169\pi$

Divide each side by $\pi$ to get rid of it on both sides. Your new equation should look like this.

r^2 = 169

Now, take the square root of each side. The square root of 169 is 13, so...

r=13

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Example Question #31 : How To Find The Length Of A Radius

True or false: A circle with diameter 25 has radius 12.5.

Possible Answers:

False

True

Correct answer:

True

Explanation:

The radius of a circle is one half (or 0.5) times its diameter, so a circle with diameter 25 has radius $25 \times 0.5 = 12.5$ .

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Basic Geometry : Circles

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #21 : How To Find The Length Of A Radius

Example Question #23 : How To Find The Length Of A Radius

Example Question #24 : How To Find The Length Of A Radius

Example Question #22 : How To Find The Length Of A Radius

Example Question #23 : How To Find The Length Of A Radius

Example Question #24 : How To Find The Length Of A Radius

Example Question #184 : Basic Geometry

Example Question #24 : How To Find The Length Of A Radius

Example Question #30 : How To Find The Length Of A Radius

Example Question #31 : How To Find The Length Of A Radius

All Basic Geometry Resources

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