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Basic Geometry : Radius

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Example Questions

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:

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.

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

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.

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

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Example Question #51 : How To Find Circumference

The rectangle and the circle share a center, $\displaystyle C$. Find the circumference of the circle.

Possible Answers:

$48\pi$ $\displaystyle 48\pi$

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

$12\pi$ $\displaystyle 12\pi$

$24\pi$ $\displaystyle 24\pi$

Correct answer:

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

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.

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

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

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

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

Simplify.

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

Reduce.

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

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

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

Recall how to find the circumference of a circle:

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

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

$\text{Circumference}=10\pi$ $\displaystyle \text{Circumference}=10\pi$

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Example Question #52 : How To Find Circumference

The rectangle and the circle share a center, $\displaystyle C$. Find the circumference of the circle.

Possible Answers:

$36\pi$ $\displaystyle 36\pi$

$4\sqrt{10}\pi$ $\displaystyle 4\sqrt{10}\pi$

The circumference of the circle cannot be determined.

$3\sqrt{17}\pi$ $\displaystyle 3\sqrt{17}\pi$

Correct answer:

$3\sqrt{17}\pi$ $\displaystyle 3\sqrt{17}\pi$

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.

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

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

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

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

Simplify.

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

Reduce.

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

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

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

Recall how to find the circumference of a circle:

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

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

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

Report an Error

Example Question #53 : How To Find Circumference

The rectangle and the circle share the the same center, $\displaystyle C$. Find the circumference of the circle.

Possible Answers:

$26\pi$ $\displaystyle 26\pi$

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

Cannot be determined

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

Correct answer:

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

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.

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

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

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

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

Simplify.

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

Reduce.

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

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

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

Recall how to find the circumference of a circle:

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

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

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

Report an Error

Example Question #54 : How To Find Circumference

The rectangle and the circle share a center, $\displaystyle C$. Find the circumference of the circle.

Possible Answers:

$3\sqrt{17}\pi$ $\displaystyle 3\sqrt{17}\pi$

$3\sqrt{15}\pi$ $\displaystyle 3\sqrt{15}\pi$

$3\sqrt{26}\pi$ $\displaystyle 3\sqrt{26}\pi$

$3\sqrt{23}\pi$ $\displaystyle 3\sqrt{23}\pi$

Correct answer:

$3\sqrt{26}\pi$ $\displaystyle 3\sqrt{26}\pi$

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.

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

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

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

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

Simplify.

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

Reduce.

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

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

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

Recall how to find the circumference of a circle:

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

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

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

Report an Error

Example Question #241 : Plane Geometry

The rectangle and the circle share a center, $\displaystyle C$. Find the circumference of the circle.

Possible Answers:

$3\sqrt{13}\pi$ $\displaystyle 3\sqrt{13}\pi$

$6\sqrt{2}\pi$ $\displaystyle 6\sqrt{2}\pi$

$3\sqrt{11}\pi$ $\displaystyle 3\sqrt{11}\pi$

$3\sqrt{10}\pi$ $\displaystyle 3\sqrt{10}\pi$

Correct answer:

$3\sqrt{10}\pi$ $\displaystyle 3\sqrt{10}\pi$

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.

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

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

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

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

Simplify.

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

Reduce.

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

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

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

Recall how to find the circumference of a circle:

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

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

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

Report an Error

Example Question #56 : How To Find Circumference

The rectangle and the circle share a center, $\displaystyle C$. Find the circumference of the circle.

Possible Answers:

The circumference of the circle cannot be determined.

$10\sqrt2$ $\displaystyle 10\sqrt2$

$\sqrt{205}\pi$ $\displaystyle \sqrt{205}\pi$

$42\sqrt{7}\pi$ $\displaystyle 42\sqrt{7}\pi$

Correct answer:

$\sqrt{205}\pi$ $\displaystyle \sqrt{205}\pi$

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.

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

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

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

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

Simplify.

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

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

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

Recall how to find the circumference of a circle:

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

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

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

Report an Error

Example Question #242 : Plane Geometry

The rectangle and the circle share a center, $\displaystyle C$. Find the circumference of the circle.

Possible Answers:

$3\sqrt{26}\pi$ $\displaystyle 3\sqrt{26}\pi$

$2\sqrt{10}\pi$ $\displaystyle 2\sqrt{10}\pi$

$\sqrt{110}\pi$ $\displaystyle \sqrt{110}\pi$

$\sqrt{109}\pi$ $\displaystyle \sqrt{109}\pi$

Correct answer:

$\sqrt{109}\pi$ $\displaystyle \sqrt{109}\pi$

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.

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

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

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

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

Simplify.

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

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

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

Recall how to find the circumference of a circle:

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

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

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

Report an Error

Example Question #58 : How To Find Circumference

The rectangle and the circle share a center, $\displaystyle C$. Find the circumference of the circle.

Possible Answers:

$\sqrt{73}\pi$ $\displaystyle \sqrt{73}\pi$

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

$24\pi$ $\displaystyle 24\pi$

$6\sqrt{3}\pi$ $\displaystyle 6\sqrt{3}\pi$

Correct answer:

$\sqrt{73}\pi$ $\displaystyle \sqrt{73}\pi$

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.

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

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

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

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

Simplify.

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

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

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

Recall how to find the circumference of a circle:

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

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

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

Report an Error

Example Question #59 : How To Find Circumference

The rectangle and the circle share a center, $\displaystyle C$. Find the circumference of the circle.

Possible Answers:

$\sqrt{130}\pi$ $\displaystyle \sqrt{130}\pi$

$33\pi$ $\displaystyle 33\pi$

$4\sqrt{7}\pi$ $\displaystyle 4\sqrt{7}\pi$

The circumference of the circle cannot be determined.

Correct answer:

$\sqrt{130}\pi$ $\displaystyle \sqrt{130}\pi$

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.

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

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

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

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

Simplify.

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

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

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

Recall how to find the circumference of a circle:

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

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

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

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Basic Geometry : Radius

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #42 : 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 #241 : Plane Geometry

Example Question #56 : How To Find Circumference

Example Question #242 : Plane Geometry

Example Question #58 : How To Find Circumference

Example Question #59 : How To Find Circumference

All Basic Geometry Resources

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