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Basic Geometry : How to find circumference

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Example Question #51 : How To Find Circumference

The rectangle and the circle share a center, . Find the circumference of the circle.

Possible Answers:

$10\pi$

$12\pi$

$48\pi$

$24\pi$

Correct answer:

$10\pi$

Explanation:

Notice that the diagonal of the rectangle is also the diameter of the circle.

Use the Pythagorean theorem to find the length of the diagonal.

$\text{length}^2+\text{width}^2=\text{diagonal}^2$

$\text{diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

Substitute in the values of the length and the width to find the length of the diagonal.

$\text{diagonal}=\sqrt{6^2+8^2}$

Simplify.

$\text{diagonal}=\sqrt{100}$

Reduce.

$\text{diagonal}=10$

Now, recall the relationship between the diagonal and the diameter.

$\text{diagonal}=\text{diameter}=10$

Recall how to find the circumference of a circle:

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

Substitute in the value of the diameter to find the circumference.

$\text{Circumference}=10\pi$

Report an Error

Example Question #51 : How To Find Circumference

The rectangle and the circle share a center, . Find the circumference of the circle.

Possible Answers:

The circumference of the circle cannot be determined.

$36\pi$

$4\sqrt{10}\pi$

$3\sqrt{17}\pi$

Correct answer:

$3\sqrt{17}\pi$

Explanation:

Notice that the diagonal of the rectangle is also the diameter of the circle.

Use the Pythagorean theorem to find the length of the diagonal.

$\text{length}^2+\text{width}^2=\text{diagonal}^2$

$\text{diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

Substitute in the values of the length and the width to find the length of the diagonal.

$\text{diagonal}=\sqrt{3^2+12^2}$

Simplify.

$\text{diagonal}=\sqrt{153}$

Reduce.

$\text{diagonal}=3\sqrt{17}$

Now, recall the relationship between the diagonal and the diameter.

$\text{diagonal}=\text{diameter}=3\sqrt{17}$

Recall how to find the circumference of a circle:

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

Substitute in the value of the diameter to find the circumference.

$\text{Circumference}=3\sqrt{17}\pi$

Report an Error

Example Question #52 : How To Find Circumference

The rectangle and the circle share the the same center, . Find the circumference of the circle.

Possible Answers:

$60\pi$

Cannot be determined

$13\pi$

$26\pi$

Correct answer:

$13\pi$

Explanation:

Notice that the diagonal of the rectangle is also the diameter of the circle.

Use the Pythagorean theorem to find the length of the diagonal.

$\text{length}^2+\text{width}^2=\text{diagonal}^2$

$\text{diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

Substitute in the values of the length and the width to find the length of the diagonal.

$\text{diagonal}=\sqrt{5^2+12^2}$

Simplify.

$\text{diagonal}=\sqrt{169}$

Reduce.

$\text{diagonal}=13$

Now, recall the relationship between the diagonal and the diameter.

$\text{diagonal}=\text{diameter}=13$

Recall how to find the circumference of a circle:

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

Substitute in the value of the diameter to find the circumference.

$\text{Circumference}=13\pi$

Report an Error

Example Question #53 : How To Find Circumference

The rectangle and the circle share a center, . Find the circumference of the circle.

Possible Answers:

$3\sqrt{26}\pi$

$3\sqrt{15}\pi$

$3\sqrt{17}\pi$

$3\sqrt{23}\pi$

Correct answer:

$3\sqrt{26}\pi$

Explanation:

Notice that the diagonal of the rectangle is also the diameter of the circle.

Use the Pythagorean theorem to find the length of the diagonal.

$\text{length}^2+\text{width}^2=\text{diagonal}^2$

$\text{diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

Substitute in the values of the length and the width to find the length of the diagonal.

$\text{diagonal}=\sqrt{3^2+15^2}$

Simplify.

$\text{diagonal}=\sqrt{234}$

Reduce.

$\text{diagonal}=3\sqrt{26}$

Now, recall the relationship between the diagonal and the diameter.

$\text{diagonal}=\text{diameter}=3\sqrt{26}$

Recall how to find the circumference of a circle:

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

Substitute in the value of the diameter to find the circumference.

$\text{Circumference}=3\sqrt{26}\pi$

Report an Error

Example Question #54 : How To Find Circumference

The rectangle and the circle share a center, . Find the circumference of the circle.

Possible Answers:

$6\sqrt{2}\pi$

$3\sqrt{11}\pi$

$3\sqrt{10}\pi$

$3\sqrt{13}\pi$

Correct answer:

$3\sqrt{10}\pi$

Explanation:

Notice that the diagonal of the rectangle is also the diameter of the circle.

Use the Pythagorean theorem to find the length of the diagonal.

$\text{length}^2+\text{width}^2=\text{diagonal}^2$

$\text{diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

Substitute in the values of the length and the width to find the length of the diagonal.

$\text{diagonal}=\sqrt{3^2+9^2}$

Simplify.

$\text{diagonal}=\sqrt{90}$

Reduce.

$\text{diagonal}=3\sqrt{10}$

Now, recall the relationship between the diagonal and the diameter.

$\text{diagonal}=\text{diameter}=3\sqrt{10}$

Recall how to find the circumference of a circle:

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

Substitute in the value of the diameter to find the circumference.

$\text{Circumference}=3\sqrt{10}\pi$

Report an Error

Example Question #51 : How To Find Circumference

The rectangle and the circle share a center, . Find the circumference of the circle.

Possible Answers:

The circumference of the circle cannot be determined.

$\sqrt{205}\pi$

$42\sqrt{7}\pi$

$10\sqrt2$

Correct answer:

$\sqrt{205}\pi$

Explanation:

Notice that the diagonal of the rectangle is also the diameter of the circle.

Use the Pythagorean theorem to find the length of the diagonal.

$\text{length}^2+\text{width}^2=\text{diagonal}^2$

$\text{diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

Substitute in the values of the length and the width to find the length of the diagonal.

$\text{diagonal}=\sqrt{3^2+14^2}$

Simplify.

$\text{diagonal}=\sqrt{205}$

Now, recall the relationship between the diagonal and the diameter.

$\text{diagonal}=\text{diameter}=\sqrt{205}$

Recall how to find the circumference of a circle:

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

Substitute in the value of the diameter to find the circumference.

$\text{Circumference}=\sqrt{205}\pi$

Report an Error

Example Question #57 : How To Find Circumference

The rectangle and the circle share a center, . Find the circumference of the circle.

Possible Answers:

$\sqrt{110}\pi$

$3\sqrt{26}\pi$

$\sqrt{109}\pi$

$2\sqrt{10}\pi$

Correct answer:

$\sqrt{109}\pi$

Explanation:

Notice that the diagonal of the rectangle is also the diameter of the circle.

Use the Pythagorean theorem to find the length of the diagonal.

$\text{length}^2+\text{width}^2=\text{diagonal}^2$

$\text{diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

Substitute in the values of the length and the width to find the length of the diagonal.

$\text{diagonal}=\sqrt{3^2+10^2}$

Simplify.

$\text{diagonal}=\sqrt{109}$

Now, recall the relationship between the diagonal and the diameter.

$\text{diagonal}=\text{diameter}=\sqrt{109}$

Recall how to find the circumference of a circle:

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

Substitute in the value of the diameter to find the circumference.

$\text{Circumference}=\sqrt{109}\pi$

Report an Error

Example Question #58 : How To Find Circumference

The rectangle and the circle share a center, . Find the circumference of the circle.

Possible Answers:

$6\sqrt{3}\pi$

$8\pi$

$\sqrt{73}\pi$

$24\pi$

Correct answer:

$\sqrt{73}\pi$

Explanation:

Notice that the diagonal of the rectangle is also the diameter of the circle.

Use the Pythagorean theorem to find the length of the diagonal.

$\text{length}^2+\text{width}^2=\text{diagonal}^2$

$\text{diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

Substitute in the values of the length and the width to find the length of the diagonal.

$\text{diagonal}=\sqrt{3^2+8^2}$

Simplify.

$\text{diagonal}=\sqrt{73}$

Now, recall the relationship between the diagonal and the diameter.

$\text{diagonal}=\text{diameter}=\sqrt{73}$

Recall how to find the circumference of a circle:

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

Substitute in the value of the diameter to find the circumference.

$\text{Circumference}=\sqrt{73}\pi$

Report an Error

Example Question #52 : How To Find Circumference

The rectangle and the circle share a center, . Find the circumference of the circle.

Possible Answers:

The circumference of the circle cannot be determined.

$\sqrt{130}\pi$

$33\pi$

$4\sqrt{7}\pi$

Correct answer:

$\sqrt{130}\pi$

Explanation:

Notice that the diagonal of the rectangle is also the diameter of the circle.

Use the Pythagorean theorem to find the length of the diagonal.

$\text{length}^2+\text{width}^2=\text{diagonal}^2$

$\text{diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

Substitute in the values of the length and the width to find the length of the diagonal.

$\text{diagonal}=\sqrt{3^2+11^2}$

Simplify.

$\text{diagonal}=\sqrt{130}$

Now, recall the relationship between the diagonal and the diameter.

$\text{diagonal}=\text{diameter}=\sqrt{130}$

Recall how to find the circumference of a circle:

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

Substitute in the value of the diameter to find the circumference.

$\text{Circumference}=\sqrt{130}\pi$

Report an Error

Example Question #55 : How To Find Circumference

The circle and the rectangle share a center, . Find the circumference of the circle.

Possible Answers:

$4\sqrt{5}\pi$

$3\sqrt{7}\pi$

$2\sqrt{10}\pi$

$10\sqrt{3}\pi$

Correct answer:

$4\sqrt{5}\pi$

Explanation:

Notice that the diagonal of the rectangle is also the diameter of the circle.

Use the Pythagorean theorem to find the length of the diagonal.

$\text{length}^2+\text{width}^2=\text{diagonal}^2$

$\text{diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

Substitute in the values of the length and the width to find the length of the diagonal.

$\text{diagonal}=\sqrt{4^2+8^2}$

Simplify.

$\text{diagonal}=\sqrt{80}$

Reduce.

$\text{diagonal}=4\sqrt{5}$

Now, recall the relationship between the diagonal and the diameter.

$\text{diagonal}=\text{diameter}=4\sqrt{5}$

Recall how to find the circumference of a circle:

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

Substitute in the value of the diameter to find the circumference.

$\text{Circumference}=4\sqrt{5}\pi$

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 #51 : How To Find Circumference

Example Question #51 : How To Find Circumference

Example Question #52 : How To Find Circumference

Example Question #53 : How To Find Circumference

Example Question #54 : How To Find Circumference

Example Question #51 : How To Find Circumference

Example Question #57 : How To Find Circumference

Example Question #58 : How To Find Circumference

Example Question #52 : How To Find Circumference

Example Question #55 : How To Find Circumference

All Basic Geometry Resources

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