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Advanced Geometry : How to find the volume of a tetrahedron

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Example Questions

Example Question #1 : 3 Dimensional Geometry

What is the volume of the following tetrahedron? Assume the figure is a regular tetrahedron.

Possible Answers:

$\4\sqrt{3}m^3$

$\frac{2\sqrt{2}}{3}m^3$

$\frac{3\sqrt{2}}{2}m^3$

$\frac{8\sqrt{2}}{3}m^3$

$\frac{\sqrt{2}}{3}m^3$

Correct answer:

$\frac{2\sqrt{2}}{3}m^3$

Explanation:

A regular tetrahedron is composed of four equilateral triangles. The formula for the volume of a regular tetrahedron is:

$V=\frac{a^3\sqrt{2}}{12}$ , where represents the length of the side.

Plugging in our values we get:

$V=\frac{(2\: m)^3\sqrt{2}}{12}$

$V=\frac{8\: m^3\sqrt{2}}{12} = \frac{2\sqrt{2}}{3}m^3$

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

Find the volume of a tetrahedron with an edge of $1\:cm$ .

Possible Answers:

$\frac{\sqrt3}{4}\:cm^3$

$\frac{\sqrt2}{12}\:cm^3$

$6\sqrt{2}\:cm^3$

$3\:cm^3$

$\frac{1}{6}\:cm^3$

Correct answer:

$\frac{\sqrt2}{12}\:cm^3$

Explanation:

Write the formula for the volume of a tetrahedron.

$V=\frac{s^3}{6\sqrt 2}$

Substitute in the length of the edge provided in the problem.

$V=\frac{(1\:cm)^3}{6\sqrt2}=\frac{1}{6\sqrt2}\:cm^3$

Rationalize the denominator.

$V= \frac{1}{6\sqrt2}\:cm\cdot \frac{\sqrt2}{\sqrt2} = \frac{\sqrt2}{12}\:cm^3$

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

Find the volume of a tetrahedron with an edge of $\sqrt2\:in$ .

Possible Answers:

$\frac{1}{3}\:in^3$

$\sqrt3\:in^3$

$\frac{2}{3}\:in^3$

$\frac{1}{6}\:in^3$

$\frac{1}{2}\:in^3$

Correct answer:

$\frac{1}{3}\:in^3$

Explanation:

Write the formula for the volume of a tetrahedron.

$V=\frac{s^3}{6\sqrt 2}$

Substitute in the length of the edge provided in the problem:

$V=\frac{(\sqrt2\:in)^3}{6\sqrt2}$

Cancel out the $\sqrt2$ in the denominator with one in the numerator:

$V= \frac{(\sqrt2)^3}{6\sqrt2}\:in^3$

A square root is being raised to the power of two in the numerator; these two operations cancel each other out. After canceling those operations, reduce the remaining fraction to arrive at the correct answer:

$V= \frac{(\sqrt2)^2}{6}\:in^3=\frac{2}{6}\:in^3=\frac{1}{3}\:in^3$

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

Find the volume of a tetrahedron with an edge of $6\:cm$ .

Possible Answers:

$18\sqrt2\:cm^3$

$36\sqrt2\:cm^3$

Correct answer:

$18\sqrt2\:cm^3$

Explanation:

Write the formula for finding the volume of a tetrahedron.

$V=\frac{s^3}{6\sqrt 2}$

Substitute in the edge length provided in the problem.

$V=\frac{(6\:cm)^3}{6\sqrt 2}=\frac{6^3}{6\sqrt 2}\:cm^3$

Cancel out the in the denominator with part of the 6^3 in the numerator:

$V= \frac{6^2}{\sqrt2}\:cm^3$

Expand, rationalize the denominator, and reduce to arrive at the correct answer:

$V= \frac{36}{\sqrt2}\:cm^3=\frac{36}{\sqrt2}\:cm^3 \cdot \frac{\sqrt2}{\sqrt2} =\frac{36\sqrt2}{2}\:cm^3 = 18\sqrt2\:cm^3$

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

Find the volume of a tetrahedron with an edge of $6\sqrt2$ .

Possible Answers:

Correct answer:

Explanation:

Write the formula the volume of a tetrahedron.

$V=\frac{s^3}{6\sqrt 2}$

Substitute the edge length provided in the equation into the formula.

$V= \frac{(6\sqrt 2)^3}{6\sqrt 2}$

Cancel out the denominator with part of the numerator and solve the remaining part of the numerator to arrive at the correct answer.

$V= (6\sqrt 2)^2 = 72$

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

Find the volume of a tetrahedron with an edge of $\sqrt5$ .

Possible Answers:

$\frac{125}{6}$

$5\sqrt3$

$\frac{125\sqrt2}{12}$

$\frac{5\sqrt3}{9}$

$\frac{25\sqrt10}{12}$

Correct answer:

$\frac{25\sqrt10}{12}$

Explanation:

Write the formula the volume of a tetrahedron and substitute in the provided edge length.

$V=\frac{s^3}{6\sqrt 2}= \frac{(\sqrt 5)^3}{6\sqrt 2} = \frac{25\sqrt 5}{6\sqrt2}$

Rationalize the denominator to arrive at the correct answer.

$\frac{25\sqrt5}{6\sqrt2} \cdot \frac{\sqrt2}{\sqrt2} = \frac{25\sqrt 10}{12}$

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

Find the volume of the regular tetrahedron with side length $5\:cm$ .

Possible Answers:

$12.03\:cm^3$

$14.73\:cm^3$

$125.00\:cm^3$

$2.95\:cm^3$

$20.83\:cm^3$

Correct answer:

$14.73\:cm^3$

Explanation:

The formula for the volume of a regular tetrahedron is:

$V = \frac{s^3}{6\sqrt{2}}$

Where is the length of side. Using this formula and the given values, we get:

$V = \frac{5\:cm^3}{6\sqrt{2}} = 14.73\:cm^3$

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

What is the volume of a regular tetrahedron with edges of 6cm ?

Possible Answers:

$\small 6\sqrt{2} cm^3$

$\small \sqrt{2} cm^3$

$\small \frac{36}{\sqrt{2}} cm^3$

$\small \frac{216}{\sqrt{2}} cm^3$

Correct answer:

$\small \frac{36}{\sqrt{2}} cm^3$

Explanation:

The volume of a tetrahedron is found with the formula:

$\small V= \frac{a^3}{6\sqrt{2}}$ ,

where $\small a$ is the length of the edges.

When $\small a=6$ ,

$\small V=\frac{36}{\sqrt{2}}$ .

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

What is the volume of a regular tetrahedron with edges of $\small \frac{2}{3} cm$ ?

Possible Answers:

$\small \frac{4}{81\sqrt{2}} cm^3$

$\small \frac{2}{9\sqrt{2}}cm^3$

$\small \frac{2}{9\sqrt{2}}cm^2$

None of the above.

$\small \frac{4}{81\sqrt{2}} cm^2$

Correct answer:

$\small \frac{4}{81\sqrt{2}} cm^3$

Explanation:

The volume of a tetrahedron is found with the formula,

$\small V= \frac{a^3}{6\sqrt{2}}$ where $\small a$ is the length of the edges.

When $\small a= \frac{2}{3}$ the volume becomes,

$\small V= \left ( \frac{2}{3} ^3\right )\div 6\sqrt{2}$

$\small V=\frac{8}{27}\div 6\sqrt{2}$

$\small V= \frac{8}{27}\times\frac{1}{6\sqrt{2}}=\frac{4}{81\sqrt{2}}$

The answer is in volume, so it must be in a cubic measurement!

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

What is the volume of a regular tetrahedron with edges of $\small \small \frac{4}{5}cm$ ?

Possible Answers:

$\small \frac{125}{\sqrt{2}}$ $\small cm^3$

$\small \frac{32}{\sqrt{2}}$ $\small cm^3$

$\small \frac{375}{32\sqrt{2}}$

$\small 375\sqrt{2}$

None of the above.

Correct answer:

None of the above.

Explanation:

The volume of a tetrahedron is found with the formula $\small V= \frac{a^3}{6\sqrt{2}}$ where $\small a$ is the length of the edges.

When $\small \small a= \frac{4}{5}$

$\small \small V= \left ( \frac{4}{5} ^3\right )\div 6\sqrt{2}$

$\small \small V=\frac{64}{125}\div 6\sqrt{2}$

$\small \small \small V= \frac{64}{125}\times\frac{1}{6\sqrt{2}}=\frac{64}{750\sqrt{2}}=\frac{32}{375\sqrt{2}}$

This answer is not found, so it is "none of the above."

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Advanced Geometry : How to find the volume of a tetrahedron

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #1 : 3 Dimensional Geometry

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

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

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

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

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

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

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

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

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

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