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SAT Math : How to find the volume of a pyramid

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

The volume of a 6-foot-tall square pyramid is 8 cubic feet. How long are the sides of the base?

Possible Answers:

$4\ ft^{2}$ $\displaystyle 4\ ft^{2}$

$4\ ft$ $\displaystyle 4\ ft$

$2\ ft^{2}$ $\displaystyle 2\ ft^{2}$

$2\ ft$ $\displaystyle 2\ ft$

$1.5\ ft$ $\displaystyle 1.5\ ft$

Correct answer:

$2\ ft$ $\displaystyle 2\ ft$

Explanation:

Volume of a pyramid is

$\frac{1}{3}\cdot (Area\ of\ the\ base)\cdot (height)$ $\displaystyle \frac{1}{3}\cdot (Area\ of\ the\ base)\cdot (height)$

Thus:

$8=\frac{1}{3}\cdot (Area\ of\ the\ base)\cdot (6)$ $\displaystyle 8=\frac{1}{3}\cdot (Area\ of\ the\ base)\cdot (6)$

$8=2\cdot (Area\ of\ the\ base)$ $\displaystyle 8=2\cdot (Area\ of\ the\ base)$

Area of the base is $4\ ft^{2}$ $\displaystyle 4\ ft^{2}$.

Therefore, each side is $2\ ft$ $\displaystyle 2\ ft$.

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

A right pyramid with a square base has a height that is twice the length of one edge of the base. If the height of the pyramid is 6 meters, find the volume of the pyramid.

Possible Answers:

Correct answer:

Explanation:

If the height, which is twice the length of the base edges, measures 6 meters, then each base edge must measure 3 meters.

Since the base is a square, the area of the base is 3 x 3 = 9.

Therefore the volume of the right pyramid is V = (1/3) x area of the base x height = 1/3(9)(6) = 18.

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Example Question #3 : How To Find The Volume Of A Pyramid

Find the volume of the pyramid shown below:

Screen shot 2015 10 27 at 3.33.57 pm

Possible Answers:

$\displaystyle 24$

689 $\displaystyle 689$

$\frac{17}{3}$ $\displaystyle \frac{17}{3}$

432 $\displaystyle 432$

144 $\displaystyle 144$

Correct answer:

144 $\displaystyle 144$

Explanation:

The formula for the area of a pyramid is $\frac{lwh}{3}$ $\displaystyle \frac{lwh}{3}$. In this case, the length is $\displaystyle 8$, the width is $\displaystyle 6$, and the height is $\displaystyle 9$.

$8\times6\times9=432$ $\displaystyle 8\times6\times9=432$ and $439\div3=144$ $\displaystyle 439\div3=144$.

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Example Question #4 : How To Find The Volume Of A Pyramid

Pyramid question

Figure not drawn to scale

In the pyramid above, the base is a square. The distance between points C and D is 6 inches and the length of side b is 5 inches. What is the volume of this pyramid?

Possible Answers:

$10\ cm^3$ $\displaystyle 10\ cm^3$

$12\ cm^3$ $\displaystyle 12\ cm^3$

$30\ cm^3$ $\displaystyle 30\ cm^3$

$24\ cm^3$ $\displaystyle 24\ cm^3$

$25\ cm^3$ $\displaystyle 25\ cm^3$

Correct answer:

$24\ cm^3$ $\displaystyle 24\ cm^3$

Explanation:

Pyramid question notes1

To find the volume of a pyramid, you need to use the equation below:

$Volume=\frac{(area\, of\, the\, base)(height)}{3}$ $\displaystyle Volume=\frac{(area\, of\, the\, base)(height)}{3}$

To find the height (shown by the yellow line), we can draw a right triangle using the yellow line, blue line and side b (5 inches). Because the hypotenuse is 5 inches, using the common Pythagorean 3-4-5 triple. The blue line is 3 inches and the yellow line (height) is 4 inches. Also, to find side a, we can use the blue line (3 inches) and half of the red line (3 inches) and the Pythagorean Theorum.

$3^2+3^2=(side\, a)^2$ $\displaystyle 3^2+3^2=(side\, a)^2$

$9+9=(side\, a)^2$ $\displaystyle 9+9=(side\, a)^2$

$18=(side\, a)^2$ $\displaystyle 18=(side\, a)^2$

$\sqrt{18}\, in=side\, a$ $\displaystyle \sqrt{18}\, in=side\, a$

Because the base is a square, the area of the base is equal to the square of side a:

$(side a)^2=18\: in^2$ $\displaystyle (side a)^2=18\: in^2$

Now we plug in these values to find the volume:

$Volume=\frac{(18)(4)}{3}$ $\displaystyle Volume=\frac{(18)(4)}{3}$

$Volume=\frac{(72)}{3}$ $\displaystyle Volume=\frac{(72)}{3}$

$Volume=24\, in^3$ $\displaystyle Volume=24\, in^3$

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Example Question #5 : How To Find The Volume Of A Pyramid

Pyramid

Calculate the volume of the rectangular pyramid with height h=4in. $\displaystyle h=4in.$, base width w=6in. $\displaystyle w=6in.$, and base length l=6in. $\displaystyle l=6in.$

Possible Answers:

48in.^3 $\displaystyle 48in.^3$

36in.^3 $\displaystyle 36in.^3$

32in.^3 $\displaystyle 32in.^3$

60in.^3 $\displaystyle 60in.^3$

72in.^3 $\displaystyle 72in.^3$

Correct answer:

48in.^3 $\displaystyle 48in.^3$

Explanation:

The volume $\displaystyle V$ of a rectangular pyramid with height $\displaystyle h$, base width $\displaystyle w$, and base length $\displaystyle l$ is given by

$V=\frac{1}{3}lwh$ $\displaystyle V=\frac{1}{3}lwh$.

For this pyramid, h=4in. $\displaystyle h=4in.$, w=6in. $\displaystyle w=6in.$, and l=6in. $\displaystyle l=6in.$ To calculate its volume, substitute the values for $\displaystyle h$, $\displaystyle w$, and $\displaystyle l$ into the formula:

$V=\frac{1}{3}(6)(6)(4)=\frac{144}{3}=48$ $\displaystyle V=\frac{1}{3}(6)(6)(4)=\frac{144}{3}=48$

Therefore, the volume of the given rectangular pyramid is 48in.^3 $\displaystyle 48in.^3$

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SAT Math : How to find the volume of a pyramid

Study concepts, example questions & explanations for SAT Math

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Example Questions

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

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

Example Question #3 : How To Find The Volume Of A Pyramid

Example Question #4 : How To Find The Volume Of A Pyramid

Example Question #5 : How To Find The Volume Of A Pyramid

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