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ISEE Upper Level Quantitative : How to find the volume of a pyramid

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Example Questions

Example Question #5 : Solid Geometry

Pyramid 1 has a square base with sidelength $\displaystyle x$; its height is $\displaystyle 2x$.

Pyramid 2 has a square base with sidelength $\displaystyle 2x$; its height is $\displaystyle x$.

Which is the greater quantity?

(a) The volume of Pyramid 1

(b) The volume of Pyramid 2

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

Correct answer:

(b) is greater.

Explanation:

Use the formula $V = \frac {1}{3} Bh$ $\displaystyle V = \frac {1}{3} Bh$on each pyramid.

(a) $B = x^{2}; h = 2x$ $\displaystyle B = x^{2}; h = 2x$

$V = \frac {1}{3} Bh = \frac {1}{3} \cdot x^{2} \cdot 2x = \frac {2}{3} x^{3}$ $\displaystyle V = \frac {1}{3} Bh = \frac {1}{3} \cdot x^{2} \cdot 2x = \frac {2}{3} x^{3}$

(b) $B = (2 x)^{2} = 4x^{2}; h = x$ $\displaystyle B = (2 x)^{2} = 4x^{2}; h = x$

$V = \frac {1}{3} Bh = \frac {1}{3} \cdot 4x^{2} \cdot x = \frac {4}{3} x^{3}$ $\displaystyle V = \frac {1}{3} Bh = \frac {1}{3} \cdot 4x^{2} \cdot x = \frac {4}{3} x^{3}$

Regardless of $\displaystyle x$, (b) is the greater quantity.

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Example Question #303 : Geometry

Which is the greater quantity?

(a) The volume of a pyramid with height 4, the base of which has sidelength 1

(b) The volume of a pyramid with height 1, the base of which has sidelength 2

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(a) and (b) are equal.

Explanation:

The volume of a pyramid with height $\displaystyle h$ and a square base with sidelength $\displaystyle s$ is

$V = \frac{1}{3} Bh = \frac{1}{3} s^{2}h$ $\displaystyle V = \frac{1}{3} Bh = \frac{1}{3} s^{2}h$.

(a) Substitute s=1,h=4 $\displaystyle s=1,h=4$: $V = \frac{1}{3} s^{2}h = \frac{1}{3} \cdot 1^{2} \cdot 4 = \frac{4}{3}$ $\displaystyle V = \frac{1}{3} s^{2}h = \frac{1}{3} \cdot 1^{2} \cdot 4 = \frac{4}{3}$

(b) Substitute s=2,h=1 $\displaystyle s=2,h=1$: $V = \frac{1}{3} s^{2}h = \frac{1}{3} \cdot 2^{2} \cdot 1 = \frac{4}{3}$ $\displaystyle V = \frac{1}{3} s^{2}h = \frac{1}{3} \cdot 2^{2} \cdot 1 = \frac{4}{3}$

The two pyramids have equal volume.

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

Which is the greater quantity?

(a) The volume of a pyramid whose base is a square with sidelength 8 inches

(b) The volume of a pyramid whose base is an equilateral triangle with sidelength one foot

Possible Answers:

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

Correct answer:

It is impossible to tell from the information given.

Explanation:

The volume of a pyramid is one-third of the product of the height and the area of the base. The areas of the bases can be calculated, but no information is given about the heights of the pyramids. There is not enough information to determine which one has the greater volume.

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Example Question #2 : Solid Geometry

A pyramid with a square base has height equal to the perimeter of its base. Its volume is $\displaystyle V$. In terms of $\displaystyle V$, what is the length of each side of its base?

Possible Answers:

$\sqrt[3]{3V }$ $\displaystyle \sqrt[3]{3V }$

$\frac{\sqrt[3]{3V }}{8}$ $\displaystyle \frac{\sqrt[3]{3V }}{8}$

$\sqrt[3]{\frac{3V}{4}}$ $\displaystyle \sqrt[3]{\frac{3V}{4}}$

$\sqrt[3]{\frac{3V}{2}}$ $\displaystyle \sqrt[3]{\frac{3V}{2}}$

$\frac{\sqrt[3]{3V }}{2}$ $\displaystyle \frac{\sqrt[3]{3V }}{2}$

Correct answer:

$\sqrt[3]{\frac{3V}{4}}$ $\displaystyle \sqrt[3]{\frac{3V}{4}}$

Explanation:

The volume of a pyramid is given by the formula

$V = \frac{1}{3} A h$ $\displaystyle V = \frac{1}{3} A h$

where $\displaystyle A$ is the area of its base and $\displaystyle h$ is its height.

Let $\displaystyle s$ be the length of one side of the square base. Then the height is equal to the perimeter of that square, so

h = 4s $\displaystyle h = 4s$

and the area of the base is

$A = s^{2}$ $\displaystyle A = s^{2}$

So the volume formula becomes

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

Solve for $\displaystyle s$:

$\frac{4}{3} \cdot s^{3} = V$ $\displaystyle \frac{4}{3} \cdot s^{3} = V$

$\frac{3}{4} \cdot \frac{4}{3} \cdot s^{3} =\frac{3}{4} \cdot V$ $\displaystyle \frac{3}{4} \cdot \frac{4}{3} \cdot s^{3} =\frac{3}{4} \cdot V$

$s^{3} =\frac{3V}{4}$ $\displaystyle s^{3} =\frac{3V}{4}$

$s =\sqrt[3]{\frac{3V}{4}}$ $\displaystyle s =\sqrt[3]{\frac{3V}{4}}$

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : How to find the volume of a pyramid

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #5 : Solid Geometry

Example Question #303 : Geometry

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

Example Question #2 : Solid Geometry

All ISEE Upper Level Quantitative Resources

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