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SSAT Upper Level Math : Volume of a Three-Dimensional Figure

Study concepts, example questions & explanations for SSAT Upper Level Math

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All SSAT Upper Level Math Resources

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Example Questions

Example Question #13 : How To Find The Volume Of A Tetrahedron

Find the volume of a regular tetrahedron with side lengths of .

Possible Answers:

$\frac{2x^3\sqrt2}{3}$

$\frac{4x^3\sqrt2}{3}$

$8x^3\sqrt2$

$\frac{3x^3\sqrt2}{4}$

Correct answer:

$\frac{2x^3\sqrt2}{3}$

Explanation:

Use the following formula to find the volume of a regular tetrahedron:

$\text{Volume}=\frac{\sqrt2}{12}side^3$

Now, plug in the given side length.

$\text{Volume}=\frac{\sqrt2}{12}(2x)^3$

$\text{Volume}=\frac{8x^3\sqrt2}{12}$

$\text{Volume}=\frac{2x^3\sqrt2}{3}$

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Example Question #14 : How To Find The Volume Of A Tetrahedron

Find the volume of a regular tetrahedron with side lengths of .

Possible Answers:

$16a^3\sqrt2$

$\frac{32a^3\sqrt2}{3}$

$\frac{16a^3\sqrt2}{3}$

$64a^3\sqrt2$

Correct answer:

$\frac{16a^3\sqrt2}{3}$

Explanation:

Use the following formula to find the volume of a regular tetrahedron:

$\text{Volume}=\frac{\sqrt2}{12}side^3$

Now, plug in the given side length.

$\text{Volume}=\frac{\sqrt2}{12}(4a)^3$

$\text{Volume}=\frac{64a^3\sqrt2}{12}$

$\text{Volume}=\frac{16a^3\sqrt2}{3}$

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Example Question #15 : How To Find The Volume Of A Tetrahedron

Find the volume of a regular tetrahedron with side lengths of 12h .

Possible Answers:

$966h^3\sqrt2$

$144h^3\sqrt2$

$1796h^3\sqrt2$

$288h^3\sqrt2$

Correct answer:

$144h^3\sqrt2$

Explanation:

Use the following formula to find the volume of a regular tetrahedron:

$\text{Volume}=\frac{\sqrt2}{12}side^3$

Now, plug in the given side length.

$\text{Volume}=\frac{\sqrt2}{12}(12h)^3$

$\text{Volume}=\frac{1728h^3\sqrt2}{12}$

$\text{Volume}=144h^3\sqrt2$

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Example Question #16 : How To Find The Volume Of A Tetrahedron

Find the volume of a regular tetrahedron with side lengths of $\frac{1}{2}$ .

Possible Answers:

$\frac{\sqrt2}{48}$

$\frac{\sqrt2}{96}$

$\frac{4\sqrt2}{3}$

$\frac{\sqrt2}{24}$

Correct answer:

$\frac{\sqrt2}{96}$

Explanation:

Use the following formula to find the volume of a regular tetrahedron:

$\text{Volume}=\frac{\sqrt2}{12}side^3$

Now, plug in the given side length.

$\text{Volume}=\frac{\sqrt2}{12}(\frac{1}{2})^3$

$\text{Volume}=\frac{\sqrt2}{12}(\frac{1}{8})$

$\text{Volume}=\frac{\sqrt2}{96}$

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

Find the volume of a regular tetrahedron with side lengths of $\frac{1}{3}$ .

Possible Answers:

$\frac{\sqrt2}{27}$

$\frac{\sqrt2}{54}$

$\frac{\sqrt2}{12}$

$\frac{\sqrt2}{324}$

Correct answer:

$\frac{\sqrt2}{324}$

Explanation:

Use the following formula to find the volume of a regular tetrahedron:

$\text{Volume}=\frac{\sqrt2}{12}side^3$

Now, plug in the given side length.

$\text{Volume}=\frac{\sqrt2}{12}(\frac{1}{3})^3$

$\text{Volume}=\frac{\sqrt2}{12}(\frac{1}{27})$

$\text{Volume}=\frac{\sqrt2}{324}$

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Example Question #5 : Volume Of A Rectangular Solid

A given rectangular prism has a length of $9\:cm$ , a width of $5\:cm$ , and a height of $7\:cm$ . What is the volume of the prism?

Possible Answers:

$315\:cm^{3}$

$63\:cm^{2}$

$315\:cm^{2}$

$45\:cm^{3}$

$63\:cm^{3}$

Correct answer:

$315\:cm^{3}$

Explanation:

The volume of a given prism V=Bh , where is the base area and is the height. For a rectangular prism, the base area $B=l \times w$ , or length times width. Therefore:

V=Bh

$V=(l\times w)h$

Substituting in our known values:

$V=(9\:cm\times 5\:cm)7\:cm$

$V=45\:cm^{2}\times 7\:cm$

$V=315\:cm^{3}$

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Example Question #1 : How To Find The Volume Of A Prism

Find the volume of a circular prism with a base radius of $11\:cm$ and a height of $20\:cm$ .

Possible Answers:

$2420\pi\:cm^{3}$

$220\pi\:cm^{3}$

$440\pi\:cm^{2}$

$2420\pi\:cm^{2}$

$440\pi\:cm^{3}$

Correct answer:

$2420\pi\:cm^{3}$

Explanation:

The volume of a given prism V=Bh , where is the base area and is the height. For a circular prism, the base area $B=\pi r^{2}$ , or pi times the square of the radius. Therefore:

$V=(\pi r^{2})h$

Substituting in our known values:

$V=(\pi (11\:cm)^{2})(20\:cm)$

$V=(121\pi\:cm^{2})(20\:cm)$

$V=2420\pi\:cm^{3}$

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Example Question #1 : How To Find The Volume Of A Prism

A triangular prism has a triangle base length of $4\:cm$ , a triangle height length of $8\:cm$ , and a prism height of $12\:cm$ . What is the volume of the prism?

Possible Answers:

$192\:cm^{2}$

$192\:cm^{3}$

Not enough information provided

$384\:cm^{3}$

$384\:cm^{2}$

Correct answer:

$192\:cm^{3}$

Explanation:

The volume of a given prism V=Bh , where is the base area and is the prism height. For a triangular prism, the base area $B=\frac{1}{2}bh_{2}$ , where is the base of the triangle and $h_{2}$ is the height of the triangle (not of the prism).

Therefore, we can substitute the base area equation into the equation for the volume of a prism:

V=Bh

$V=(\frac{1}{2}bh_{2})h$

Substituting in our known values:

$V=(\frac{1}{2}(4\:cm)(8\:cm))(12\:cm)$

$V=(16\:cm^{2})(12\:cm)$

$V=192\:cm^{3}$

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Example Question #61 : Volume

A given rectangular prism has a length of , a width of , and a height of . What is the volume of the prism?

Possible Answers:

572

574

576

Correct answer:

576

Explanation:

The volume of a rectangular prism is the product of its base area and its height : V=Ah . Since we can determine the base area of the rectangular prism from its length and width, we can rewrite the equation and solve:

V=Ah

V=lwh

V=(12)(6)(8)

$V=12 \times48$

V=576

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Example Question #2 : Volume Of A Rectangular Solid

A given rectangular prism has a length of 10cm , a width of 7cm , and a height of 15cm , what is its volume?

Possible Answers:

$1050cm^{3}$

$32cm^{3}$

$525cm^{3}$

$85cm^{3}$

$1050cm^{2}$

Correct answer:

$1050cm^{3}$

Explanation:

V=Ah

V=lwh

V=(10)(7)(25)

V=1050

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All SSAT Upper Level Math Resources

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SSAT Upper Level Math : Volume of a Three-Dimensional Figure

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #13 : How To Find The Volume Of A Tetrahedron

Example Question #14 : How To Find The Volume Of A Tetrahedron

Example Question #15 : How To Find The Volume Of A Tetrahedron

Example Question #16 : How To Find The Volume Of A Tetrahedron

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

Example Question #5 : Volume Of A Rectangular Solid

Example Question #1 : How To Find The Volume Of A Prism

Example Question #1 : How To Find The Volume Of A Prism

Example Question #61 : Volume

Example Question #2 : Volume Of A Rectangular Solid

All SSAT Upper Level Math Resources

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