Basic Geometry : How to find the length of the diameter

Study concepts, example questions & explanations for Basic Geometry

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Example Questions

Example Question #341 : Circles

Find the length of the diameter of a circle inscribed in a square that has a diagonal of \(\displaystyle 10\).

Possible Answers:

\(\displaystyle 5\sqrt2\)

\(\displaystyle 5\)

\(\displaystyle 25\pi\)

\(\displaystyle 10\pi\)

Correct answer:

\(\displaystyle 5\sqrt2\)

Explanation:

13

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.

\(\displaystyle \text{Diagonal}^2=\text{side}^2+\text{side}^2\)

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

\(\displaystyle \text{side}^2=\frac{\text{Diagonal}^2}{2}\)

\(\displaystyle \text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}\)

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

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

Simplify.

\(\displaystyle \text{side}=5\sqrt2\)

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

\(\displaystyle \text{diameter}=\text{side}=5\sqrt2\)

Example Question #342 : Plane Geometry

Find the length of the diameter of a circle inscribed in a square that has a diagonal of \(\displaystyle 20\).

Possible Answers:

\(\displaystyle 5\sqrt2\)

\(\displaystyle 50\)

\(\displaystyle 200\)

\(\displaystyle 10\sqrt2\)

Correct answer:

\(\displaystyle 10\sqrt2\)

Explanation:

13

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.

\(\displaystyle \text{Diagonal}^2=\text{side}^2+\text{side}^2\)

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

\(\displaystyle \text{side}^2=\frac{\text{Diagonal}^2}{2}\)

\(\displaystyle \text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}\)

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

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

Simplify.

\(\displaystyle \text{side}=10\sqrt2\)

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

\(\displaystyle \text{diameter}=\text{side}=10\sqrt2\)

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 \(\displaystyle 24\).

Possible Answers:

\(\displaystyle 6\sqrt2\)

\(\displaystyle 18\pi\)

\(\displaystyle 12\sqrt2\)

\(\displaystyle 24\pi\)

Correct answer:

\(\displaystyle 12\sqrt2\)

Explanation:

13

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.

\(\displaystyle \text{Diagonal}^2=\text{side}^2+\text{side}^2\)

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

\(\displaystyle \text{side}^2=\frac{\text{Diagonal}^2}{2}\)

\(\displaystyle \text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}\)

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

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

Simplify.

\(\displaystyle \text{side}=12\sqrt2\)

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

\(\displaystyle \text{diameter}=\text{side}=12\sqrt2\)

Example Question #344 : Circles

Find the length of the diameter of a circle inscribed in a square that has a diagonal of \(\displaystyle 26\).

Possible Answers:

\(\displaystyle 26\pi\)

\(\displaystyle 14\sqrt2\)

\(\displaystyle 13\sqrt2\)

\(\displaystyle \frac{13}{2}\sqrt2\)

Correct answer:

\(\displaystyle 13\sqrt2\)

Explanation:

13

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.

\(\displaystyle \text{Diagonal}^2=\text{side}^2+\text{side}^2\)

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

\(\displaystyle \text{side}^2=\frac{\text{Diagonal}^2}{2}\)

\(\displaystyle \text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}\)

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

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

Simplify.

\(\displaystyle \text{side}=13\sqrt2\)

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

\(\displaystyle \text{diameter}=\text{side}=13\sqrt2\)

Example Question #341 : Plane Geometry

Find the length of the diameter of a circle inscribed in a square that has a diagonal of \(\displaystyle 36\).

Possible Answers:

\(\displaystyle 18\sqrt2\)

\(\displaystyle 18\)

\(\displaystyle 54\sqrt2\)

\(\displaystyle 72\)

Correct answer:

\(\displaystyle 18\sqrt2\)

Explanation:

13

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.

\(\displaystyle \text{Diagonal}^2=\text{side}^2+\text{side}^2\)

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

\(\displaystyle \text{side}^2=\frac{\text{Diagonal}^2}{2}\)

\(\displaystyle \text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}\)

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

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

Simplify.

\(\displaystyle \text{side}=18\sqrt2\)

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

\(\displaystyle \text{diameter}=\text{side}=18\sqrt2\) 

Example Question #351 : Circles

Find the length of the diameter of a circle inscribed in a square that has a diagonal of \(\displaystyle 48\).

Possible Answers:

\(\displaystyle 24\sqrt2\)

\(\displaystyle 12\sqrt2\)

\(\displaystyle 48\sqrt2\)

\(\displaystyle 6\sqrt2\)

Correct answer:

\(\displaystyle 24\sqrt2\)

Explanation:

13

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.

\(\displaystyle \text{Diagonal}^2=\text{side}^2+\text{side}^2\)

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

\(\displaystyle \text{side}^2=\frac{\text{Diagonal}^2}{2}\)

\(\displaystyle \text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}\)

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

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

Simplify.

\(\displaystyle \text{side}=24\sqrt2\)

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

\(\displaystyle \text{diameter}=\text{side}=24\sqrt2\)

Example Question #351 : Plane Geometry

Find the length of the diameter given the area is \(\displaystyle 16\pi\).

Possible Answers:

\(\displaystyle 16\)

\(\displaystyle 12\)

\(\displaystyle 8\)

\(\displaystyle 4\)

Correct answer:

\(\displaystyle 8\)

Explanation:

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

\(\displaystyle A=\pi{r^2}\Rightarrow r=\sqrt{\frac{A}{\pi}}\)

\(\displaystyle r=\sqrt{\frac{16\pi}{\pi}}=\sqrt{16}=4\)

\(\displaystyle d=2r=2*4=8\)

Example Question #353 : Circles

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

Possible Answers:

\(\displaystyle 60 units\)

\(\displaystyle 70 units\)

\(\displaystyle 40 units\)

\(\displaystyle 90 units\)

\(\displaystyle 80 units\)

Correct answer:

\(\displaystyle 80 units\)

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.

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

Then we plug in numbers and solve. 

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

\(\displaystyle 125.7 = \pi (r)\)

\(\displaystyle 40 = radius\)

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

\(\displaystyle Diameter = 2(Radius)\)

Therefore, the diameter is equal to \(\displaystyle 80 units\)

Example Question #352 : Plane Geometry

Find the length of the diameter given radius is 7.

Possible Answers:

\(\displaystyle 14\pi\)

\(\displaystyle 49\pi\)

\(\displaystyle 14\)

\(\displaystyle 28\pi\)

Correct answer:

\(\displaystyle 14\)

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,

\(\displaystyle \\d=2r\\d=2*7\\d=14\).

Example Question #353 : Plane Geometry

Find the length of the diameter of a circle given an area of  \(\displaystyle 9\pi\).

Possible Answers:

\(\displaystyle 6\pi\)

\(\displaystyle 6\)

\(\displaystyle 9\)

\(\displaystyle 3\)

Correct answer:

\(\displaystyle 6\)

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,

\(\displaystyle A=\pi{r^2}\Rightarrow r=\sqrt{\frac{A}{\pi}}\)

\(\displaystyle r=\sqrt{\frac{9\pi}{\pi}}=\sqrt9=3\)

\(\displaystyle d=2r=2*3=6\)

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