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

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Example Question #1 : Pyramids

A square pyramid has a volume of 363cm^3 and a height of $\small 9cm$ . What is the perimeter of the base of the pyramid?

Possible Answers:

36cm

44cm

80cm

100cm

121cm

Correct answer:

44cm

Explanation:

The formula for the volume of a pyramid is:

$\small V=\frac{lwh}{3}$

We know that V=363 and h=9 ; however, this still leaves us with two variables in the equation: and . By definition, a square pyramid's base has sides of equal length, meaning that and are the same. Therefore, we can substitute $\small lw$ for $\small l\cdot l$ , or $\small l^2$ .

This gives us a new equation of:

$\small V=\frac{l^2h}{3}$

We then plug in the variables we know:

$\small 363=\frac{9l^2}{3}$

Multiply both sides by 3:

$\small 1089=9l^2$

Divide both sides by 9:

l^2=121

l=11

Therefore, we now know that the length of the pyramid's base is $\small 11cm$ . The question, however, asks for the perimeter of the pyramid's base. Since all of the sides of the base are the same, they must all be $\small 11cm$ . So we multiply $11cm\cdot 4=44cm$ . Therefore, the perimeter of the pyramid's base is $\small 44cm$ .

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

A square pyramid has a slant height of 8. The edge of a face is 12. What is the measure of an edge of its base?

Possible Answers:

15.6

8.3

8.9

9.0

17.8

Correct answer:

17.8

Explanation:

In this kind of a problem, it's helpful to draw out all information that's given.

Pyramid

Because the triangular faces of a pyramid are isosceles triangles, the slant height can be seen to divide it into two right triangles. The goal is to find the measure of the edge of the square base. This quickly becomes a problem revolving around the use of the Pythagorean theorem.

If we take the measures 12 and 8 to be the hypotenuse and one of the legs, respectively, we can use the Pythagoream theorem to solve for the "base leg." Then, realizing that this measure if only half of the the total length of the square base edge, that value must be multiplied by 2.

a^2+b^2=c^2

a^2+8^2=12^2

a^2+64=144

a^2=80

$a=\sqrt{80}$

$a={\color{Green} 8.9}$

Therefore, 2a=17.8

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

Study concepts, example questions & explanations for ACT Math

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Example Question #1 : Pyramids

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

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