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SSAT Upper Level Math : How to find the volume of a polyhedron

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Example Questions

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

Find the volume of a square pyramid that has a height of $\displaystyle 9$ and a side length of $\displaystyle 4$.

Possible Answers:

$\displaystyle 42$

$\displaystyle 48$

$\displaystyle 60$

$\displaystyle 54$

Correct answer:

$\displaystyle 48$

Explanation:

The formula to find the volume of a square pyramid is

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

So plugging in the information given from the question,

$\text{Volume}=4^2\frac{9}{3}=16\times3=48$ $\displaystyle \text{Volume}=4^2\frac{9}{3}=16\times3=48$

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

Find the volume of a square pyramid with a height of $\displaystyle 12$ and a length of a side of its square base of $\displaystyle 3$.

Possible Answers:

$\displaystyle 24$

$\displaystyle 60$

$\displaystyle 36$

$\displaystyle 48$

Correct answer:

$\displaystyle 36$

Explanation:

The formula to find the volume of a square pyramid is

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

So plugging in the information given from the question,

$\text{Volume}=(3^2)\frac{12}{3}=9\times 4=36$ $\displaystyle \text{Volume}=(3^2)\frac{12}{3}=9\times 4=36$

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Example Question #2 : How To Find The Volume Of A Polyhedron

Find the volume of a regular hexagonal prism that has a height of $\displaystyle 10$. The side length of the hexagon base is $\displaystyle 12$.

Possible Answers:

$660\sqrt3$ $\displaystyle 660\sqrt3$

$4320\sqrt3$ $\displaystyle 4320\sqrt3$

$2160\sqrt3$ $\displaystyle 2160\sqrt3$

$1440\sqrt3$ $\displaystyle 1440\sqrt3$

Correct answer:

$2160\sqrt3$ $\displaystyle 2160\sqrt3$

Explanation:

The formula to find the volume of a hexagonal prism is

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

Plugging in the values given by the question will give

$\text{Volume}=\frac{3\sqrt3}{2}(12^2)(10)=2160\sqrt3$ $\displaystyle \text{Volume}=\frac{3\sqrt3}{2}(12^2)(10)=2160\sqrt3$

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Example Question #25 : Volume Of A Three Dimensional Figure

In terms of $\displaystyle x$, find the volume of a regular hexagonal prism that has a height of $\displaystyle 5$. The hexagon base has side lengths of $\displaystyle 4x$.

Possible Answers:

$240x\sqrt3$ $\displaystyle 240x\sqrt3$

$120x\sqrt3$ $\displaystyle 120x\sqrt3$

$120x^2\sqrt3$ $\displaystyle 120x^2\sqrt3$

$360x^2\sqrt3$ $\displaystyle 360x^2\sqrt3$

Correct answer:

$120x^2\sqrt3$ $\displaystyle 120x^2\sqrt3$

Explanation:

The formula to find the volume of a hexagonal prism is

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

Plugging in the values given by the question will give

$\text{Volume}=\frac{3\sqrt3}{2}(4x)^2(5)=120x^2\sqrt3$ $\displaystyle \text{Volume}=\frac{3\sqrt3}{2}(4x)^2(5)=120x^2\sqrt3$

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Example Question #3 : How To Find The Volume Of A Polyhedron

Find the volume of a regular tetrahedron with a side length of $\displaystyle 12$.

Possible Answers:

$144\sqrt2$ $\displaystyle 144\sqrt2$

$1728\sqrt2$ $\displaystyle 1728\sqrt2$

$288\sqrt2$ $\displaystyle 288\sqrt2$

$966\sqrt2$ $\displaystyle 966\sqrt2$

Correct answer:

$144\sqrt2$ $\displaystyle 144\sqrt2$

Explanation:

The formula to find the volume of a tetrahedron is

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

Plugging in the information given by the question gives

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

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

Find the volume of a regular octahedron that has a side length of $\displaystyle 8$.

Possible Answers:

$100\sqrt2$ $\displaystyle 100\sqrt2$

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

$512\sqrt2$ $\displaystyle 512\sqrt2$

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

Correct answer:

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

Explanation:

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

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

Plugging in the information from the question,

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

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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 $\displaystyle 6y$.

Possible Answers:

$144y^3\sqrt2$ $\displaystyle 144y^3\sqrt2$

$216y^3\sqrt2$ $\displaystyle 216y^3\sqrt2$

$72y^3\sqrt2$ $\displaystyle 72y^3\sqrt2$

$84y^3\sqrt2$ $\displaystyle 84y^3\sqrt2$

Correct answer:

$72y^3\sqrt2$ $\displaystyle 72y^3\sqrt2$

Explanation:

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

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

Plugging in the information from the question,

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

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

Find the volume of a regular hexahedron with a side length of $\displaystyle 14$.

Possible Answers:

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

196 $\displaystyle 196$

$1372\sqrt3$ $\displaystyle 1372\sqrt3$

2744 $\displaystyle 2744$

Correct answer:

2744 $\displaystyle 2744$

Explanation:

A regular hexahedron is another name for a cube.

To find the volume of a cube,

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

Plugging in the information given in the question gives

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

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

Find the volume of a regular octahedron with side lengths of $\displaystyle 9t$.

Possible Answers:

$243t\sqrt2$ $\displaystyle 243t\sqrt2$

$243t^3\sqrt2$ $\displaystyle 243t^3\sqrt2$

$729t^3\sqrt2$ $\displaystyle 729t^3\sqrt2$

$81t^3\sqrt2$ $\displaystyle 81t^3\sqrt2$

Correct answer:

$243t^3\sqrt2$ $\displaystyle 243t^3\sqrt2$

Explanation:

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

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

Plugging in the information from the question,

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

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

Find the volume of a prism that has a right triangle base with leg lengths of $\displaystyle 3$ and $\displaystyle 8$ and a height of $\displaystyle 12$.

Possible Answers:

432 $\displaystyle 432$

$\displaystyle 96$

288 $\displaystyle 288$

144 $\displaystyle 144$

Correct answer:

144 $\displaystyle 144$

Explanation:

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

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

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

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SSAT Upper Level Math : How to find the volume of a polyhedron

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

All SSAT Upper Level Math Resources

Example Questions

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

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

Example Question #2 : How To Find The Volume Of A Polyhedron

Example Question #25 : Volume Of A Three Dimensional Figure

Example Question #3 : How To Find The Volume Of A Polyhedron

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

All SSAT Upper Level Math Resources

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