How to find the volume of a polyhedron - SSAT Upper Level Quantitative (Math)

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Question

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

Answer

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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Question

Find the volume of a square pyramid that has a height of and a side length of .

Answer

The formula to find the volume of a square pyramid is

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

So plugging in the information given from the question,

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

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Question

Find the volume of a square pyramid with a height of and a length of a side of its square base of .

Answer

The formula to find the volume of a square pyramid is

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

So plugging in the information given from the question,

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

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Question

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

Answer

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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Question

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

Answer

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

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

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Question

In terms of , find the volume of a regular hexagonal prism that has a height of . The hexagon base has side lengths of .

Answer

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

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

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Question

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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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