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ISEE Upper Level Math : How to find the length of an edge of a tetrahedron

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Example Questions

Example Question #1 : Tetrahedrons

A triangular pyramid, or tetrahedron, with volume 100 has four vertices with Cartesian coordinates

$\displaystyle (0,0,0), (n,0, 0), (0,10,0), (0, 0, 5)$

where n > 0 $\displaystyle n > 0$.

Evaluate $\displaystyle n$.

Possible Answers:

n = 18 $\displaystyle n = 18$

n = 12 $\displaystyle n = 12$

n = 9 $\displaystyle n = 9$

n = 24 $\displaystyle n = 24$

n = 6 $\displaystyle n = 6$

Correct answer:

n = 12 $\displaystyle n = 12$

Explanation:

The tetrahedron is as follows (figure not to scale):

Tetrahedron

This is a triangular pyramid with a right triangle with legs 10 and $\displaystyle n$ as its base; the area of the base is

$B = \frac{1}{2} \cdot 10 \cdot n = 5n$ $\displaystyle B = \frac{1}{2} \cdot 10 \cdot n = 5n$

The height of the pyramid is 5, so

$V = \frac{1}{3} \cdot 5n \cdot 5 = \frac{25n}{3}$ $\displaystyle V = \frac{1}{3} \cdot 5n \cdot 5 = \frac{25n}{3}$

Set this equal to 100 to get $\displaystyle n$:

$\frac{25n}{3} = 100$ $\displaystyle \frac{25n}{3} = 100$

$n = 100 \cdot \frac{3}{25} = 12$ $\displaystyle n = 100 \cdot \frac{3}{25} = 12$

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Example Question #1 : Tetrahedrons

A triangular pyramid, or tetrahedron, with volume 1,000 has four vertices with Cartesian coordinates

$\displaystyle (0,0,0), (n,0, 0), (0,n,0), (0, 0, n)$

where n > 0 $\displaystyle n > 0$.

Evaluate $\displaystyle n$.

Possible Answers:

$n = 10 \sqrt[3]{2}$ $\displaystyle n = 10 \sqrt[3]{2}$

$n = 10 \sqrt[3]{6}$ $\displaystyle n = 10 \sqrt[3]{6}$

$n = 10 \sqrt[3]{3}$ $\displaystyle n = 10 \sqrt[3]{3}$

n = 10 $\displaystyle n = 10$

$n = 10 \sqrt[3]{12}$ $\displaystyle n = 10 \sqrt[3]{12}$

Correct answer:

$n = 10 \sqrt[3]{6}$ $\displaystyle n = 10 \sqrt[3]{6}$

Explanation:

The tetrahedron is as follows:

Pyramid

This is a triangular pyramid with a right triangle with two legs of measure $\displaystyle n$ as its base; the area of the base is

$B = \frac{1}{2} \cdot n \cdot n = \frac{n^{2}}{2}$ $\displaystyle B = \frac{1}{2} \cdot n \cdot n = \frac{n^{2}}{2}$

Since the height of the pyramid is also $\displaystyle n$, the volume is

$V = \frac{1}{3} \cdot \frac{n^{2}}{2} \cdot n = \frac{n^{3}}{6}$ $\displaystyle V = \frac{1}{3} \cdot \frac{n^{2}}{2} \cdot n = \frac{n^{3}}{6}$.

Set this equal to 1,000:

$\frac{n^{3}}{6} = 1,000$ $\displaystyle \frac{n^{3}}{6} = 1,000$

$n^{3} = 6,000$ $\displaystyle n^{3} = 6,000$

$n = \sqrt[3]{6,000} = \sqrt[3]{1,000} \cdot \sqrt[3]{6} = 10 \sqrt[3]{6}$ $\displaystyle n = \sqrt[3]{6,000} = \sqrt[3]{1,000} \cdot \sqrt[3]{6} = 10 \sqrt[3]{6}$

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Example Question #1 : How To Find The Length Of An Edge Of A Tetrahedron

A triangular pyramid, or tetrahedron, with volume 240 has four vertices with Cartesian coordinates

$\displaystyle (0,0,0), (n,0, 0), (0,n,0), (0, 0, 24)$

where n > 0 $\displaystyle n > 0$.

Evaluate $\displaystyle n$.

Possible Answers:

$n = 2 \sqrt{30}$ $\displaystyle n = 2 \sqrt{30}$

$n = \sqrt{30}$ $\displaystyle n = \sqrt{30}$

$n = 2 \sqrt{15}$ $\displaystyle n = 2 \sqrt{15}$

$n= 6\sqrt{10}$ $\displaystyle n= 6\sqrt{10}$

$n= 3\sqrt{10}$ $\displaystyle n= 3\sqrt{10}$

Correct answer:

$n = 2 \sqrt{15}$ $\displaystyle n = 2 \sqrt{15}$

Explanation:

The tetrahedron is as follows (figure not to scale):

Tetrahedron

This is a triangular pyramid with a right triangle with two legs of measure $\displaystyle n$ as its base; the area of the base is

$B = \frac{1}{2} \cdot n \cdot n = \frac{n^{2}}{2}$ $\displaystyle B = \frac{1}{2} \cdot n \cdot n = \frac{n^{2}}{2}$

The height of the pyramid is 24, so the volume is

$V = \frac{1}{3} \cdot \frac{n^{2}}{2} \cdot 24= 4 n^{2}$ $\displaystyle V = \frac{1}{3} \cdot \frac{n^{2}}{2} \cdot 24= 4 n^{2}$

Set this equal to 240 to get $\displaystyle n$:

$4 n^{2} = 240$ $\displaystyle 4 n^{2} = 240$

$n^{2} = 60$ $\displaystyle n^{2} = 60$

$n = \sqrt{60} = \sqrt{4} \cdot \sqrt{15} = 2 \sqrt{15}$ $\displaystyle n = \sqrt{60} = \sqrt{4} \cdot \sqrt{15} = 2 \sqrt{15}$

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : How to find the length of an edge of a tetrahedron

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : Tetrahedrons

Example Question #1 : Tetrahedrons

Example Question #1 : How To Find The Length Of An Edge Of A Tetrahedron

All ISEE Upper Level Math Resources

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