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Advanced Geometry : How to find the surface area of a tetrahedron

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Example Questions

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

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

Tetrahedron

Possible Answers:

$8\sqrt{3}m^2$

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

$3\sqrt{3}m^2$

$2\sqrt{3}m^2$

$4\sqrt{3}m^2$

Correct answer:

$4\sqrt{3}m^2$

Explanation:

A tetrahedron is a three-dimensonal figure where each side is an equilateral triangle. Therefore, each angle in the triangle is $60^{\circ}$ .

In the figure, we know the value of the side $(2\: m)$ and the value of the base $(1\: m)$ . Since dividing the triangle by half creates a $30^{\circ}-60^{\circ}-90^{\circ}$ triangle, we know the value of must be $\sqrt{3}\: m$ .

Therefore, the area of one side of the tetrahedron is:

$A = (b)(h) = (1\: m)(\sqrt{3}\: m) = \sqrt{3}\: m^2$

Since there are four sides of a tetrahedron, the surface area is:

$SA = 4 (b)(h) = 4\sqrt{3}\: m^2$

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

A regular tetrahedron has side lengths s=1 . What is the surface area of the described solid?

Possible Answers:

$4\sqrt{3}$

$2\sqrt3$

$\sqrt{3}$

Correct answer:

$\sqrt{3}$

Explanation:

The area of one face of the triangle can be found either through trigonometry or the Pythagorean Theorem.

Since all the sides of the triangle are , the height is then $\frac{\sqrt{3}}{2}$ , so the area of each face is:

$\frac{bh}{2}=\frac{1}{2} \cdot 1 \cdot \frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{4}$

There are four faces, so the area of the tetrahedron is:

$4 \cdot \frac{bh}{2}=\sqrt3$

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

Find the surface area of a regular tetrahedron with a side length of .

Possible Answers:

$4\sqrt3$

$2\sqrt3$

$8\sqrt3$

$16\sqrt3$

Correct answer:

$16\sqrt3$

Explanation:

Use the following formula to find the surface area of a regular tetrahedron.

$\text{Surface Area}=\sqrt3 (side^2)$

Now, substitute in the value of the side length into the equation.

$\text{Surface Area}=\sqrt3(4^2)=16\sqrt3$

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

Find the surface area of a regular tetrahedron with a side length of .

Possible Answers:

$12\sqrt3$

$36\sqrt3$

$144\sqrt3$

$96\sqrt3$

Correct answer:

$144\sqrt3$

Explanation:

Use the following formula to find the surface area of a regular tetrahedron.

$\text{Surface Area}=\sqrt3 (side^2)$

Now, substitute in the value of the side length into the equation.

$\text{Surface Area}=\sqrt3(12^2)=144\sqrt3$

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

In terms of , find the surface area of a regular tetrahedron that has a side length of .

Possible Answers:

$5x^2\sqrt3$

$25x^2\sqrt3$

$25x\sqrt3$

$125x^2\sqrt3$

Correct answer:

$25x^2\sqrt3$

Explanation:

Use the following formula to find the surface area of a regular tetrahedron.

$\text{Surface Area}=\sqrt3 (side^2)$

Now, substitute in the value of the side length into the equation.

$\text{Surface Area}=\sqrt3(5x)^2=25x^2\sqrt3$

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

In terms of , find the surface area of a regular tetrahedron with side lengths of .

Possible Answers:

$3q\sqrt3$

$9q\sqrt3$

$27q^2\sqrt3$

$9q^2\sqrt3$

Correct answer:

$9q^2\sqrt3$

Explanation:

Use the following formula to find the surface area of a regular tetrahedron.

$\text{Surface Area}=\sqrt3 (side^2)$

Now, substitute in the value of the side length into the equation.

$\text{Surface Area}=\sqrt3(3q)^2=9q^2\sqrt3$

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Example Question #5 : How To Find The Surface Area Of A Tetrahedron

The surface area of a regular tetrahedron is $36\sqrt3$ . If the length of each side is , find the value of .

Possible Answers:

Correct answer:

Explanation:

Use the following formula to find the surface area of a regular tetrahedron.

$\text{Surface Area}=\sqrt3 (side^2)$

Now, substitute in the value of the side length into the equation.

$36\sqrt3=\sqrt3(3a)^2$

Now, solve for .

9a^2=36

a^2=4

a=2

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Example Question #6 : How To Find The Surface Area Of A Tetrahedron

The surface area of a regular tetrahedron is . If each side length is , find the value of . Round to the nearest tenths place.

Possible Answers:

0.9

8.4

1.7

2.4

Correct answer:

0.9

Explanation:

Use the following formula to find the surface area of a regular tetrahedron.

$\text{Surface Area}=\sqrt3 (side^2)$

Now, substitute in the value of the side length into the equation.

$54=\sqrt3(6x)^2$

Solve for .

$36x^2\sqrt3=54$

$x^2\sqrt3=1.5$

$x^2=\frac{1.5}{\sqrt3}=0.866$

$x=\sqrt{0.866}=0.9$

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Example Question #3 : How To Find The Surface Area Of A Tetrahedron

The surface area of a regular tetrahedron is . If each side length is , find the value of . Round to the nearest tenths place.

Possible Answers:

2.3

1.9

1.5

0.7

Correct answer:

1.5

Explanation:

Use the following formula to find the surface area of a regular tetrahedron.

$\text{Surface Area}=\sqrt3 (side^2)$

Now, substitute in the value of the side length into the equation.

$64=\sqrt3(4x)^2$

$16x^2=\frac{64}{\sqrt3}$

$x^2=\frac{64}{16\sqrt3}=2.309$

x=1.5

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

The surface area of a regular tetrahedron is $100\sqrt3$ . If each side length is x-1 , find the value of .

Possible Answers:

Correct answer:

Explanation:

Use the following formula to find the surface area of a regular tetrahedron.

$\text{Surface Area}=\sqrt3 (side^2)$

Now, substitute in the value of the side length into the equation and solve for .

$100\sqrt3=\sqrt3(x-1)^2$

(x-1)^2=100

x^2-2x+1=100

x^2-2x-99=0

(x-11)(x+9)=0

x=11, x=-9

Since we are dealing with a 3-dimensional shape, only x=11 is valid.

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Advanced Geometry : How to find the surface area of a tetrahedron

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

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

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

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

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

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

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

Example Question #5 : How To Find The Surface Area Of A Tetrahedron

Example Question #6 : How To Find The Surface Area Of A Tetrahedron

Example Question #3 : How To Find The Surface Area Of A Tetrahedron

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

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