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

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All SSAT Upper Level Math Resources

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Example Questions

Example Question #4 : How To Find The Volume Of A Polyhedron

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

Possible Answers:

$966\sqrt2$

$1728\sqrt2$

$288\sqrt2$

$144\sqrt2$

Correct answer:

$144\sqrt2$

Explanation:

The formula to find the volume of a tetrahedron is

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

Plugging in the information given by the question gives

$\text{Volume}=\frac{\sqrt2}{12}(12)^3=144\sqrt2$

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Example Question #5 : How To Find The Volume Of A Polyhedron

Find the volume of a regular octahedron that has a side length of .

Possible Answers:

$100\sqrt2$

$\frac{512\sqrt2}{3}$

$\frac{500\sqrt2}{3}$

$512\sqrt2$

Correct answer:

$\frac{512\sqrt2}{3}$

Explanation:

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

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

Plugging in the information from the question,

$\text{Volume}=\frac{\sqrt2}{3}(8)^3=\frac{512\sqrt2}{3}$

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Example Question #6 : How To Find The Volume Of A Polyhedron

Find the volume of a regular octahedron that has a side length of .

Possible Answers:

$216y^3\sqrt2$

$144y^3\sqrt2$

$72y^3\sqrt2$

$84y^3\sqrt2$

Correct answer:

$72y^3\sqrt2$

Explanation:

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

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

Plugging in the information from the question,

$\text{Volume}=\frac{\sqrt2}{3}(6y)^3=72y^3\sqrt2$

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Example Question #7 : How To Find The Volume Of A Polyhedron

Find the volume of a regular hexahedron with a side length of .

Possible Answers:

2744

$\frac{2744\sqrt2}{3}$

$1372\sqrt3$

196

Correct answer:

2744

Explanation:

A regular hexahedron is another name for a cube.

To find the volume of a cube,

$\text{Volume}=(side)^3$

Plugging in the information given in the question gives

$\text{Volume}=(14)^3=2744$

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Example Question #8 : How To Find The Volume Of A Polyhedron

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

Possible Answers:

$243t\sqrt2$

$243t^3\sqrt2$

$81t^3\sqrt2$

$729t^3\sqrt2$

Correct answer:

$243t^3\sqrt2$

Explanation:

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

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

Plugging in the information from the question,

$\text{Volume}=\frac{\sqrt2}{3}(9t)^3=243t^3\sqrt2$

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Example Question #9 : How To Find The Volume Of A Polyhedron

Find the volume of a prism that has a right triangle base with leg lengths of and and a height of .

Possible Answers:

432

144

288

Correct answer:

144

Explanation:

To find the volume of a prism, multiply the area of the base by the height.

$\text{Area of Base}=\frac{3\times8}{2}=12$

$\text{Volume}=12\times height=12\times12=144$

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Example Question #31 : Volume Of A Three Dimensional Figure

Find the volume of a regular hexagonal prism that has a height of . The side length of the hexagon base is .

Possible Answers:

$150\sqrt3$

$300\sqrt3$

$10\sqrt3$

$\frac{20\sqrt3}{3}$

Correct answer:

$150\sqrt3$

Explanation:

The formula to find the volume of a hexagonal prism is

$\text{Volume}=\frac{3\sqrt3}{2}(side)^2(height)$

Plugging in the values given by the question will give

$\text{Volume}=\frac{3\sqrt3}{2}(5^2)(4)=150\sqrt3$

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Example Question #32 : Volume Of A Three Dimensional Figure

Find the volume of a regular tetrahedron that has side lengths of .

Possible Answers:

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

$12a^3\sqrt2$

$\frac{9a^3\sqrt2}{4}$

$3a^3\sqrt2$

Correct answer:

$\frac{9a^3\sqrt2}{4}$

Explanation:

The formula to find the volume of a tetrahedron is

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

Plugging in the information given by the question gives

$\text{Volume}=\frac{\sqrt2}{12}(3a)^3=\frac{27a^3\sqrt2}{12}=\frac{9a^3\sqrt2}{4}$

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

In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates (0,0,0), (10,0,0), (5, 18, 0), (8,8,8) .

Give its volume.

Possible Answers:

720

360

120

160

240

Correct answer:

240

Explanation:

A tetrahedron is a triangular pyramid and can be looked at as such.

Three of the vertices - (0,0,0), (10,0,0), (5, 18, 0) - are on the -plane, and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles:

Tetrahedron

Its base is 10 and its height is 18, so its area is

$B = \frac{1}{2} \cdot 10 \cdot 18 = 90$

The fourth vertex is off the -plane; its perpendicular distance to the aforementioned face is its -coordinate, 8, so this is the height of the pyramid. The volume of the pyramid is

$V = \frac{1}{3} \cdot 90 \cdot 8 = 240$

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

In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates (0,0,0), (15,0,0), (0,15,0), (0,0, 15) .

What is the volume of this tetrahedron?

Possible Answers:

$562 \frac{1}{2}$

The correct answer is not among the other responses.

$843 \frac{3}{4}$

375

$281\frac{1}{4}$

Correct answer:

$562 \frac{1}{2}$

Explanation:

The tetrahedron looks like this:

Tetrahedron

is the origin and A,B,C are the other three points, which are fifteen units away from the origin on each of the three (perpendicular) axes.

This is a triangular pyramid, and we can consider $\Delta OCB$ the base; its area is half the product of its legs, or

$B = \frac{1}{2} \cdot 15 \cdot 15 = 112\frac{1}{2}$ .

The volume of the tetrahedron is one third the product of its base and its height, the latter of which is 15. Therefore,

$V = \frac{1}{3} \cdot 15 \cdot 112\frac{1}{2} = 562 \frac{1}{2}$ .

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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 #4 : How To Find The Volume Of A Polyhedron

Example Question #5 : How To Find The Volume Of A Polyhedron

Example Question #6 : How To Find The Volume Of A Polyhedron

Example Question #7 : How To Find The Volume Of A Polyhedron

Example Question #8 : How To Find The Volume Of A Polyhedron

Example Question #9 : How To Find The Volume Of A Polyhedron

Example Question #31 : Volume Of A Three Dimensional Figure

Example Question #32 : Volume Of A Three Dimensional Figure

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

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

All SSAT Upper Level Math Resources

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