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Basic Geometry : How to find circumference

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Example Questions

Example Question #232 : Plane Geometry

Find the circumference of a circle that is inscribed in a square with a diagonal of $30\sqrt2$ $\displaystyle 30\sqrt2$ .

Possible Answers:

$60\pi$ $\displaystyle 60\pi$

$15\pi$ $\displaystyle 15\pi$

$45\pi$ $\displaystyle 45\pi$

$30\pi$ $\displaystyle 30\pi$

Correct answer:

$30\pi$ $\displaystyle 30\pi$

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.

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

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

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

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

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

$\text{side}=\frac{30\sqrt2(\sqrt2)}{2}$ $\displaystyle \text{side}=\frac{30\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=30$ $\displaystyle \text{side}=30$

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

$\text{diameter}=\text{side}=30$ $\displaystyle \text{diameter}=\text{side}=30$

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\pi\times\text{diameter}$ $\displaystyle \text{Circumference}=\pi\times\text{diameter}$

Substitute in the diameter you just found to find the circumference.

$\text{Circumference}=30\pi$ $\displaystyle \text{Circumference}=30\pi$

Report an Error

Example Question #41 : How To Find Circumference

Find the circumference of a circle inscribed in a square that has a diagonal of $32\sqrt2$ $\displaystyle 32\sqrt2$ .

Possible Answers:

$8\pi$ $\displaystyle 8\pi$

$16\pi$ $\displaystyle 16\pi$

$64\pi$ $\displaystyle 64\pi$

$32\pi$ $\displaystyle 32\pi$

Correct answer:

$32\pi$ $\displaystyle 32\pi$

Explanation:

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

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

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

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

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

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

$\text{side}=\frac{32\sqrt2(\sqrt2)}{2}$ $\displaystyle \text{side}=\frac{32\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=32$ $\displaystyle \text{side}=32$

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

$\text{diameter}=\text{side}=32$ $\displaystyle \text{diameter}=\text{side}=32$

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\pi\times\text{diameter}$ $\displaystyle \text{Circumference}=\pi\times\text{diameter}$

Substitute in the diameter you just found to find the circumference.

$\text{Circumference}=32\pi$ $\displaystyle \text{Circumference}=32\pi$

Report an Error

Example Question #234 : Plane Geometry

Find the circumference of a circle inscribed in a square that has a diagonal of $3\sqrt2$ $\displaystyle 3\sqrt2$ .

Possible Answers:

$\frac{3}{2}\pi$ $\displaystyle \frac{3}{2}\pi$

$9\pi$ $\displaystyle 9\pi$

$\frac{9}{4}\pi$ $\displaystyle \frac{9}{4}\pi$

$3\pi$ $\displaystyle 3\pi$

Correct answer:

$3\pi$ $\displaystyle 3\pi$

Explanation:

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

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

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

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

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

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

$\text{side}=\frac{3\sqrt2(\sqrt2)}{2}$ $\displaystyle \text{side}=\frac{3\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=3$ $\displaystyle \text{side}=3$

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

$\text{diameter}=\text{side}=3$ $\displaystyle \text{diameter}=\text{side}=3$

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\pi\times\text{diameter}$ $\displaystyle \text{Circumference}=\pi\times\text{diameter}$

Substitute in the diameter you just found to find the circumference.

$\text{Circumference}=3\pi$ $\displaystyle \text{Circumference}=3\pi$

Report an Error

Example Question #41 : How To Find Circumference

Find the circumference of a circle inscribed in a square that has a diagonal of $5\sqrt2$ $\displaystyle 5\sqrt2$ .

Possible Answers:

$25\pi$ $\displaystyle 25\pi$

$\frac{5}{2}\pi$ $\displaystyle \frac{5}{2}\pi$

$10\pi$ $\displaystyle 10\pi$

$5\pi$ $\displaystyle 5\pi$

Correct answer:

$5\pi$ $\displaystyle 5\pi$

Explanation:

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

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

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

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

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

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

$\text{side}=\frac{5\sqrt2(\sqrt2)}{2}$ $\displaystyle \text{side}=\frac{5\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=5$ $\displaystyle \text{side}=5$

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

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

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\pi\times\text{diameter}$ $\displaystyle \text{Circumference}=\pi\times\text{diameter}$

Substitute in the diameter you just found to find the circumference.

$\text{Circumference}=5\pi$ $\displaystyle \text{Circumference}=5\pi$

Report an Error

Example Question #236 : Plane Geometry

Find the circumference of a circle inscribed in a square that has a diagonal of $7\sqrt2$ $\displaystyle 7\sqrt2$ .

Possible Answers:

Cannot be determined

$\frac{49}{4}\pi$ $\displaystyle \frac{49}{4}\pi$

$\frac{7}{2}\pi$ $\displaystyle \frac{7}{2}\pi$

$7\pi$ $\displaystyle 7\pi$

Correct answer:

$7\pi$ $\displaystyle 7\pi$

Explanation:

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

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

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

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

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

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

$\text{side}=\frac{7\sqrt2(\sqrt2)}{2}$ $\displaystyle \text{side}=\frac{7\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=7$ $\displaystyle \text{side}=7$

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

$\text{diameter}=\text{side}=7$ $\displaystyle \text{diameter}=\text{side}=7$

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\pi\times\text{diameter}$ $\displaystyle \text{Circumference}=\pi\times\text{diameter}$

Substitute in the diameter you just found to find the circumference.

$\text{Circumference}=7\pi$ $\displaystyle \text{Circumference}=7\pi$

Report an Error

Example Question #237 : Plane Geometry

Find the circumference of a circle inscribed in a square that has a diagonal of $9\sqrt2$ $\displaystyle 9\sqrt2$ .

Possible Answers:

$81\pi$ $\displaystyle 81\pi$

$9\pi$ $\displaystyle 9\pi$

$\frac{81}{2}\pi$ $\displaystyle \frac{81}{2}\pi$

$\frac{9}{2}\pi$ $\displaystyle \frac{9}{2}\pi$

Correct answer:

$9\pi$ $\displaystyle 9\pi$

Explanation:

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

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

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

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

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

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

$\text{side}=\frac{9\sqrt2(\sqrt2)}{2}$ $\displaystyle \text{side}=\frac{9\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=9$ $\displaystyle \text{side}=9$

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

$\text{diameter}=\text{side}=9$ $\displaystyle \text{diameter}=\text{side}=9$

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\pi\times\text{diameter}$ $\displaystyle \text{Circumference}=\pi\times\text{diameter}$

Substitute in the diameter you just found to find the circumference.

$\text{Circumference}=9\pi$ $\displaystyle \text{Circumference}=9\pi$

Report an Error

Example Question #238 : Plane Geometry

Find the circumference of a circle inscribed in a square that has a diagonal of $11\sqrt2$ $\displaystyle 11\sqrt2$ .

Possible Answers:

$11\pi$ $\displaystyle 11\pi$

$\frac{11}{2}\pi$ $\displaystyle \frac{11}{2}\pi$

$121\pi$ $\displaystyle 121\pi$

$\frac{11}{4}\pi$ $\displaystyle \frac{11}{4}\pi$

Correct answer:

$11\pi$ $\displaystyle 11\pi$

Explanation:

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

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

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

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

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

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

$\text{side}=\frac{11\sqrt2(\sqrt2)}{2}$ $\displaystyle \text{side}=\frac{11\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=11$ $\displaystyle \text{side}=11$

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

$\text{diameter}=\text{side}=11$ $\displaystyle \text{diameter}=\text{side}=11$

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\pi\times\text{diameter}$ $\displaystyle \text{Circumference}=\pi\times\text{diameter}$

Substitute in the diameter you just found to find the circumference.

$\text{Circumference}=11\pi$ $\displaystyle \text{Circumference}=11\pi$

Report an Error

Example Question #239 : Plane Geometry

Find the circumference of a circle inscribed in a square that has a diagonal of $13\sqrt2$ $\displaystyle 13\sqrt2$ .

Possible Answers:

$13\pi$ $\displaystyle 13\pi$

$15\pi$ $\displaystyle 15\pi$

$\frac{169}{4}\pi$ $\displaystyle \frac{169}{4}\pi$

$\frac{13}{2}\pi$ $\displaystyle \frac{13}{2}\pi$

Correct answer:

$13\pi$ $\displaystyle 13\pi$

Explanation:

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

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

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

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

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

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

$\text{side}=\frac{13\sqrt2(\sqrt2)}{2}$ $\displaystyle \text{side}=\frac{13\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=13$ $\displaystyle \text{side}=13$

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

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

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\pi\times\text{diameter}$ $\displaystyle \text{Circumference}=\pi\times\text{diameter}$

Substitute in the diameter you just found to find the circumference.

$\text{Circumference}=13\pi$ $\displaystyle \text{Circumference}=13\pi$

Report an Error

Example Question #240 : Plane Geometry

Find the circumference of a circle inscribed in a square that has a diagonal of $\displaystyle 14$ .

Possible Answers:

$98\pi$ $\displaystyle 98\pi$

$28\pi$ $\displaystyle 28\pi$

$(14\sqrt2)\pi$ $\displaystyle (14\sqrt2)\pi$

$(7\sqrt2)\pi$ $\displaystyle (7\sqrt2)\pi$

Correct answer:

$(7\sqrt2)\pi$ $\displaystyle (7\sqrt2)\pi$

Explanation:

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

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

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

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

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

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

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

Simplify.

$\text{side}=7\sqrt2$ $\displaystyle \text{side}=7\sqrt2$

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

$\text{diameter}=\text{side}=7\sqrt2$ $\displaystyle \text{diameter}=\text{side}=7\sqrt2$

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\pi\times\text{diameter}$ $\displaystyle \text{Circumference}=\pi\times\text{diameter}$

Substitute in the diameter you just found to find the circumference.

$\text{Circumference}=(7\sqrt2)\pi$ $\displaystyle \text{Circumference}=(7\sqrt2)\pi$

Report an Error

Example Question #42 : How To Find Circumference

Find the circumference of a circle inscribed in a square that has a diagonal of $17\sqrt2$ $\displaystyle 17\sqrt2$ .

Possible Answers:

$17\pi$ $\displaystyle 17\pi$

$(17\sqrt2)\pi$ $\displaystyle (17\sqrt2)\pi$

$41\pi$ $\displaystyle 41\pi$

$34\pi$ $\displaystyle 34\pi$

Correct answer:

$17\pi$ $\displaystyle 17\pi$

Explanation:

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

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

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

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

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

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

$\text{side}=\frac{17\sqrt2(\sqrt2)}{2}$ $\displaystyle \text{side}=\frac{17\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=17$ $\displaystyle \text{side}=17$

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

$\text{diameter}=\text{side}=17$ $\displaystyle \text{diameter}=\text{side}=17$

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\pi\times\text{diameter}$ $\displaystyle \text{Circumference}=\pi\times\text{diameter}$

Substitute in the diameter you just found to find the circumference.

$\text{Circumference}=17\pi$ $\displaystyle \text{Circumference}=17\pi$

Report an Error

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Basic Geometry : How to find circumference

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #232 : Plane Geometry

Example Question #41 : How To Find Circumference

Example Question #234 : Plane Geometry

Example Question #41 : How To Find Circumference

Example Question #236 : Plane Geometry

Example Question #237 : Plane Geometry

Example Question #238 : Plane Geometry

Example Question #239 : Plane Geometry

Example Question #240 : Plane Geometry

Example Question #42 : How To Find Circumference

All Basic Geometry Resources

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