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Intermediate Geometry : Intermediate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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8 Diagnostic Tests 250 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #11 : How To Find The Volume Of A Sphere

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is $\displaystyle 6$, what is the volume of the figure?

Possible Answers:

446.13 $\displaystyle 446.13$

452.39 $\displaystyle 452.39$

458.87 $\displaystyle 458.87$

421.20 $\displaystyle 421.20$

Correct answer:

452.39 $\displaystyle 452.39$

Explanation:

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by $\displaystyle 2$.

$\text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$ $\displaystyle \text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\text{Volume of Figure}=\frac{2}{3}\pi(6)^3=144\pi=452.39$ $\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi(6)^3=144\pi=452.39$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #242 : Solid Geometry

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is $\displaystyle 7$, what is the volume of the figure?

Possible Answers:

710.02 $\displaystyle 710.02$

729.06 $\displaystyle 729.06$

718.38 $\displaystyle 718.38$

691.32 $\displaystyle 691.32$

Correct answer:

718.38 $\displaystyle 718.38$

Explanation:

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by $\displaystyle 2$.

$\text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$ $\displaystyle \text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\text{Volume of Figure}=\frac{2}{3}\pi(7)^3=\frac{686}{3}\pi=718.38$ $\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi(7)^3=\frac{686}{3}\pi=718.38$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #11 : Spheres

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is $\displaystyle 8$, what is the volume of the figure?

Possible Answers:

1120.30 $\displaystyle 1120.30$

1056.08 $\displaystyle 1056.08$

1098.67 $\displaystyle 1098.67$

1072.33 $\displaystyle 1072.33$

Correct answer:

1072.33 $\displaystyle 1072.33$

Explanation:

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by $\displaystyle 2$.

$\text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$ $\displaystyle \text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\text{Volume of Figure}=\frac{2}{3}\pi(8)^3=\frac{1024}{3}\pi=1072.33$ $\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi(8)^3=\frac{1024}{3}\pi=1072.33$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #11 : How To Find The Volume Of A Sphere

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is $\frac{1}{3}$ $\displaystyle \frac{1}{3}$, what is the volume of the figure?

Possible Answers:

0.06 $\displaystyle 0.06$

0.07 $\displaystyle 0.07$

0.08 $\displaystyle 0.08$

0.09 $\displaystyle 0.09$

Correct answer:

0.08 $\displaystyle 0.08$

Explanation:

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by $\displaystyle 2$.

$\text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$ $\displaystyle \text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\text{Volume of Figure}=\frac{2}{3}\pi(\frac{1}{3})^3=\frac{2}{81}\pi=0.08$ $\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi(\frac{1}{3})^3=\frac{2}{81}\pi=0.08$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #12 : Spheres

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is $\displaystyle 9$, what is the volume of the figure?

Possible Answers:

1526.81 $\displaystyle 1526.81$

1613.25 $\displaystyle 1613.25$

1589.67 $\displaystyle 1589.67$

1540.20 $\displaystyle 1540.20$

Correct answer:

1526.81 $\displaystyle 1526.81$

Explanation:

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by $\displaystyle 2$.

$\text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$ $\displaystyle \text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\text{Volume of Figure}=\frac{2}{3}\pi(9)^3=486\pi=1526.81$ $\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi(9)^3=486\pi=1526.81$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #11 : How To Find The Volume Of A Sphere

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is $\displaystyle 10$, what is the volume of the figure?

Possible Answers:

2086.20 $\displaystyle 2086.20$

2032.70 $\displaystyle 2032.70$

2094.40 $\displaystyle 2094.40$

2100.03 $\displaystyle 2100.03$

Correct answer:

2094.40 $\displaystyle 2094.40$

Explanation:

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by $\displaystyle 2$.

$\text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$ $\displaystyle \text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\text{Volume of Figure}=\frac{2}{3}\pi(10)^3=\frac{2000}{3}\pi=2094.40$ $\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi(10)^3=\frac{2000}{3}\pi=2094.40$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #17 : How To Find The Volume Of A Sphere

A sphere is cut in half as shown by the figure below.

If the radius of the sphere is $\displaystyle 11$, what is the volume of the figure?

Possible Answers:

2787.64 $\displaystyle 2787.64$

2558.95 $\displaystyle 2558.95$

2633.32 $\displaystyle 2633.32$

2704.05 $\displaystyle 2704.05$

Correct answer:

2787.64 $\displaystyle 2787.64$

Explanation:

Recall how to find the volume of a sphere:

$\text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Sphere}=\frac{4}{3}\pi\times\text{radius}^3$

Now since we only have half a sphere, divide the volume by $\displaystyle 2$.

$\text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$ $\displaystyle \text{Volume of Figure}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius})^2$

$\text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the given radius to find the volume of the figure.

$\text{Volume of Figure}=\frac{2}{3}\pi(11)^3=\frac{2662}{3}\pi=2787.64$ $\displaystyle \text{Volume of Figure}=\frac{2}{3}\pi(11)^3=\frac{2662}{3}\pi=2787.64$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #11 : Spheres

A sphere is cut in half and is then placed on top of a cylinder so they both share a base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

1824.50 $\displaystyle 1824.50$

1811.21 $\displaystyle 1811.21$

1809.56 $\displaystyle 1809.56$

1804.26 $\displaystyle 1804.26$

Correct answer:

1809.56 $\displaystyle 1809.56$

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the cylinder and of the half sphere.

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$ $\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

$\text{Volume of Cylinder}=\pi\times 6^2 \times 12=432 \pi$ $\displaystyle \text{Volume of Cylinder}=\pi\times 6^2 \times 12=432 \pi$

Next, find the volume of the half sphere.

$\text{Volume of Half sphere}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius}^3)=\frac{2}{3}\pi\times\text{radius}^3$ $\displaystyle \text{Volume of Half sphere}=\frac{1}{2}(\frac{4}{3}\pi\times\text{radius}^3)=\frac{2}{3}\pi\times\text{radius}^3$

Plug in the radius to find the volume of the half sphere.

$\text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 6^3=144\pi$ $\displaystyle \text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 6^3=144\pi$

Next, add up the two volumes together to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Half Sphere}+\text{Volume of Cylinder}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Half Sphere}+\text{Volume of Cylinder}$

$\text{Volume of Figure}=432\pi + 144\pi=576\pi=1809.56$ $\displaystyle \text{Volume of Figure}=432\pi + 144\pi=576\pi=1809.56$

Remember to round to $\displaystyle 2$ places after the decimal.

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Example Question #251 : Solid Geometry

A sphere is cut in half and is placed on top of a cylinder so that they share the same base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

3190.21 $\displaystyle 3190.21$

3219.59 $\displaystyle 3219.59$

3027.45 $\displaystyle 3027.45$

3058.29 $\displaystyle 3058.29$

Correct answer:

3027.45 $\displaystyle 3027.45$

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the cylinder and of the half sphere.

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$ $\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

$\text{Volume of Cylinder}=\pi\times 7^2 \times 15=735\pi$ $\displaystyle \text{Volume of Cylinder}=\pi\times 7^2 \times 15=735\pi$

Next, find the volume of the half sphere.

Plug in the radius to find the volume of the half sphere.

$\text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 7^3=\frac{686}{3}\pi$ $\displaystyle \text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 7^3=\frac{686}{3}\pi$

Next, add up the two volumes together to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$

$\text{Volume of Figure}=735\pi + \frac{686}{3}\pi=3027.45$ $\displaystyle \text{Volume of Figure}=735\pi + \frac{686}{3}\pi=3027.45$

Remember to round to $\displaystyle 2$ places after the decimal.

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Example Question #12 : How To Find The Volume Of A Sphere

A sphere is cut in half and placed on top of a cylinder so that they share the same base as shown by the figure below.

Find the volume of the figure.

Possible Answers:

590.24 $\displaystyle 590.24$

536.17 $\displaystyle 536.17$

526.89 $\displaystyle 526.89$

544.12 $\displaystyle 544.12$

Correct answer:

536.17 $\displaystyle 536.17$

Explanation:

In order to find the volume of the figure, we will first need to find the volumes of the cylinder and of the half sphere.

Recall how to find the volume of a cylinder:

$\text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$ $\displaystyle \text{Volume of Cylinder}=\pi\times\text{radius}^2\times\text{height}$

$\text{Volume of Cylinder}=\pi\times 4^2 \times 8=128\pi$ $\displaystyle \text{Volume of Cylinder}=\pi\times 4^2 \times 8=128\pi$

Next, find the volume of the half sphere.

Plug in the radius to find the volume of the half sphere.

$\text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 4^3=\frac{128}{3}\pi$ $\displaystyle \text{Volume of Half Sphere}=\frac{2}{3}\times\pi\times 4^3=\frac{128}{3}\pi$

Next, add up the two volumes together to find the volume of the figure.

$\text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$ $\displaystyle \text{Volume of Figure}=\text{Volume of Cylinder}+\text{Volume of Half Sphere}$

$\text{Volume of Figure}=128\pi + \frac{128}{3}\pi=536.17$ $\displaystyle \text{Volume of Figure}=128\pi + \frac{128}{3}\pi=536.17$

Remember to round to $\displaystyle 2$ places after the decimal.

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Intermediate Geometry : Intermediate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #11 : How To Find The Volume Of A Sphere

Example Question #242 : Solid Geometry

Example Question #11 : Spheres

Example Question #11 : How To Find The Volume Of A Sphere

Example Question #12 : Spheres

Example Question #11 : How To Find The Volume Of A Sphere

Example Question #17 : How To Find The Volume Of A Sphere

Example Question #11 : Spheres

Example Question #251 : Solid Geometry

Example Question #12 : How To Find The Volume Of A Sphere

All Intermediate Geometry Resources

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