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Basic Geometry : Diameter

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Example Question #41 : Diameter

Find the length of the diameter of a circle inscribed in a square that has a diagonal of .

Possible Answers:

$5\sqrt2$

$10\pi$

$25\pi$

Correct answer:

$5\sqrt2$

Explanation:

When you draw out the circle that is inscribed in a square, you should notice two things. The first thing you should notice is that the diagonal of the square is also the hypotenuse of a right isosceles triangle that has the side lengths of the square as its legs. The second thing you should notice is that the diameter of the circle has the same length as the length of one side of the square.

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

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

Substitute in the length of the diagonal to find the length of the square.

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

Simplify.

$\text{side}=5\sqrt2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=5\sqrt2$

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Example Question #41 : Diameter

Find the length of the diameter of a circle inscribed in a square that has a diagonal of .

Possible Answers:

$10\sqrt2$

$5\sqrt2$

200

Correct answer:

$10\sqrt2$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

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

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{20(\sqrt2)}{2}$

Simplify.

$\text{side}=10\sqrt2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=10\sqrt2$

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Example Question #42 : Diameter

Find the length of the diameter of a circle inscribed in a square that has a diagonal of .

Possible Answers:

$24\pi$

$6\sqrt2$

$18\pi$

$12\sqrt2$

Correct answer:

$12\sqrt2$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

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

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{24(\sqrt2)}{2}$

Simplify.

$\text{side}=12\sqrt2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=12\sqrt2$

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Example Question #41 : How To Find The Length Of The Diameter

Find the length of the diameter of a circle inscribed in a square that has a diagonal of .

Possible Answers:

$13\sqrt2$

$26\pi$

$14\sqrt2$

$\frac{13}{2}\sqrt2$

Correct answer:

$13\sqrt2$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

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

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{26(\sqrt2)}{2}$

Simplify.

$\text{side}=13\sqrt2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=13\sqrt2$

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Example Question #44 : Diameter

Find the length of the diameter of a circle inscribed in a square that has a diagonal of .

Possible Answers:

$18\sqrt2$

$54\sqrt2$

Correct answer:

$18\sqrt2$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

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

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{36(\sqrt2)}{2}$

Simplify.

$\text{side}=18\sqrt2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=18\sqrt2$

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Example Question #45 : Diameter

Find the length of the diameter of a circle inscribed in a square that has a diagonal of .

Possible Answers:

$48\sqrt2$

$12\sqrt2$

$24\sqrt2$

$6\sqrt2$

Correct answer:

$24\sqrt2$

Explanation:

First, use the Pythagorean theorem to find the length of a side of the square.

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

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

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

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

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{48(\sqrt2)}{2}$

Simplify.

$\text{side}=24\sqrt2$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=24\sqrt2$

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Example Question #41 : Diameter

Find the length of the diameter given the area is $16\pi$ .

Possible Answers:

Correct answer:

Explanation:

To solve, simply use the formula for area to solve for the radius and multiply by 2.

$A=\pi{r^2}\Rightarrow r=\sqrt{\frac{A}{\pi}}$

$r=\sqrt{\frac{16\pi}{\pi}}=\sqrt{16}=4$

d=2r=2*4=8

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Example Question #42 : How To Find The Length Of The Diameter

The circumference of a circle is equal to 251.33 units . Find the length of the diameter of this circle.

Possible Answers:

40 units

60 units

80 units

70 units

90 units

Correct answer:

80 units

Explanation:

There are two steps to finding the answer to this question. The first is to find the radius through the formula for the circumference of a circle.

$Area = 2 (\pi) (r)$

Then we plug in numbers and solve.

$251.33 = 2 (\pi) (r)$

$125.7 = \pi (r)$

40 = radius

Now that we have found the radius, we solve for diameter by doubling it, since..

Diameter = 2(Radius)

Therefore, the diameter is equal to 80 units .

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Example Question #42 : Diameter

Find the length of the diameter given radius is 7.

Possible Answers:

$28\pi$

$49\pi$

$14\pi$

Correct answer:

Explanation:

To solve, simply use the formula for the diameter of a circle.

Recall that the diameter of a circle is twice the radius of the circle.

Given that the radius is seven, multiply it by 2 to solve for the diameter.

Thus,

$\\d=2r\\d=2*7\\d=14$ .

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Example Question #43 : Diameter

Find the length of the diameter of a circle given an area of $9\pi$ .

Possible Answers:

$6\pi$

Correct answer:

Explanation:

To solve, simply use the formula for the area to find the length of the radius and then multiply that by 2 to find the diameter. Thus,

$A=\pi{r^2}\Rightarrow r=\sqrt{\frac{A}{\pi}}$

$r=\sqrt{\frac{9\pi}{\pi}}=\sqrt9=3$

d=2r=2*3=6

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Basic Geometry : Diameter

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #41 : Diameter

Example Question #41 : Diameter

Example Question #42 : Diameter

Example Question #41 : How To Find The Length Of The Diameter

Example Question #44 : Diameter

Example Question #45 : Diameter

Example Question #41 : Diameter

Example Question #42 : How To Find The Length Of The Diameter

Example Question #42 : Diameter

Example Question #43 : Diameter

All Basic Geometry Resources

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