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ACT Math : How to find the length of an edge of a tetrahedron

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

A regular tetrahedron has a surface area of $684cm^{2}$ . Each face of the tetrahedron has a height of 18cm . What is the length of the base of one of the faces?

Possible Answers:

24cm

17cm

19cm

76cm

38cm

Correct answer:

19cm

Explanation:

A regular tetrahedron has 4 triangular faces. The area of one of these faces is given by:

$\small A=\frac{1}{2}bh$

Because the surface area is the area of all 4 faces combined, in order to find the area for one of the faces only, we must divide the surface area by 4. We know that the surface area is $\small 684cm^2$ , therefore:

$\small A=\frac{S.A.}{4}$

$A=\frac{684cm^{2}}{4}=171cm^2$

Since we now have the area of one face, and we know the height of one face is 18cm , we can now plug these values into the original formula:

$\small A=\frac{1}{2}bh$

$\small 171=\frac{1}{2}b(18)$

$\small 171=9b$

$\small b=19$

Therefore, the length of the base of one face is $\small 19cm$ .

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

What is the length of an edge of a regular tetrahedron if its surface area is 156?

Possible Answers:

90.1

9.4

7.2

11.4

9.5

Correct answer:

9.5

Explanation:

The only given information is the surface area of the regular tetrahedron.

This is a quick problem that can be easily solved for by using the formula for the surface area of a tetrahedron:

$SA= \sqrt{3}\cdot a^2$

If we substitute in the given infomation, we are left with the edge being the only unknown.

$156 = \sqrt{3} \cdot a^2$

$\frac{156}{\sqrt{3}}=a^2$

$\sqrt{\frac{156}{\sqrt{3}}}=a$

$a=9.49 \approx{\color{Blue} 9.5}$

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Example Question #1 : Tetrahedrons

What is the length of a regular tetrahedron if one face has an area of 43.3 squared units and a slant height of $\frac{10\sqrt3}{2}$ ?

Possible Answers:

7.2

Cannot be determined

86.6

Correct answer:

Explanation:

The problem provides the information for the slant height and the area of one of the equilateral triangle faces.

The slant height merely refers to the height of this equilateral triangle.

Therefore, if we're given the area of a triangle and it's height, we should be able to solve for it's base. The base in this case will equate to the measurement of the edge. It's helpful to remember that in this case, because all faces are equilateral triangles, the measure of one length will equate to the length of all other edges.

We can use the equation that will allow us to solve for the area of a triangle:

$A=\frac{1}{2} \cdot b\cdot h$

where is base length and is height.

Substituting in the information that's been provided, we get:

$43.3 = \frac{1}{2} \cdot b \cdot \frac{10\sqrt{3}}{2}$

$b \cdot \frac{10\sqrt{3}}{2}= 2 \cdot 43.3$

$b = 2 \cdot 43.3 \cdot \frac{2}{10\sqrt{3}}$

$b =9.99971\approx {\color{Blue} 10}$

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Example Question #2 : How To Find The Length Of An Edge Of A Tetrahedron

The volume of a regular tetrahedron is 94.8. What is the measurement of one of its edges?

Possible Answers:

8.7

10.1

Cannot be determined

9.3

9.5

Correct answer:

9.3

Explanation:

This becomes a quick problem by just utilizing the formula for the volume of a tetrahedron.

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

Upon substituting the value for the volume into the formula, we are left with , which represents the edge length.

$94.8= \frac{a^3}{6\sqrt{2}}$

$a^3=94.8 \cdot 6\sqrt{2}$

$a= \sqrt[3]{94.8 \cdot 6\sqrt{2}}$

$a= {\color{Blue} 9.3}$

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

A tetrahedron has a volume that is twice the surface area times the edge. What is the length of the edge? (In the answer choices, represents edge.)

Possible Answers:

$a= \sqrt[4]{\frac{V}{\sqrt{3a^2}}}$

$a= \frac{V}{2\sqrt a^2}$

$a= \sqrt[3]{\frac{V}{2\sqrt a^2}}$

$a= \sqrt[3]{\frac{V}{2\sqrt{3}}}$

$a= \sqrt{\frac{V}{2\sqrt{3}}}$

Correct answer:

$a= \sqrt[3]{\frac{V}{2\sqrt{3}}}$

Explanation:

The problem states that the volume is:

$V= 2(\sqrt{3}\cdot a^2) \cdot a$

The point of the problem is to solve for the length of the edge. Becasuse there are no numbers, the final answer will be an expression.

In order to solve for it, we will have to rearrange the formula for volume in terms of .

$V = 2\sqrt{3}\cdot a^3$

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

$a= \sqrt[3]{\frac{V}{2\sqrt3}}$

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ACT Math : How to find the length of an edge of a tetrahedron

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

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

Example Question #1 : Tetrahedrons

Example Question #2 : How To Find The Length Of An Edge Of A Tetrahedron

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

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