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ISEE Upper Level Quantitative : Pyramids

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

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All ISEE Upper Level Quantitative Resources

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Example Questions

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

Pyramid 1 has a square base with sidelength ; its height is .

Pyramid 2 has a square base with sidelength ; its height is .

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.

(a) is greater.

(b) is greater.

(a) and (b) are equal.

Correct answer:

(b) is greater.

Explanation:

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

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

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

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

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

Regardless of , (b) is the greater quantity.

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

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:

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

The volume of a pyramid with height and a square base with sidelength is

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

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

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

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) and (b) are equal.

(b) is greater.

(a) 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 #4 : How To Find The Volume Of A Pyramid

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

Possible Answers:

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

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

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

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

$\sqrt[3]{3V }$

Correct answer:

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

Explanation:

The volume of a pyramid is given by the formula

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

where is the area of its base and is its height.

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

h = 4s

and the area of the base is

$A = s^{2}$

So the volume formula becomes

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

Solve for :

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

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

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

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

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

A pyramid with a square base has height equal to the perimeter of its base. Which is the greater quantity?

(A) Twice the area of its base

(B) The area of one of its triangular faces

Possible Answers:

It is impossible to determine which is greater from the information given

(A) and (B) are equal

(B) is greater

(A) is greater

Correct answer:

(B) is greater

Explanation:

Since the answer is not dependent on the actual dimensions, for the sake of simplicity, we assume that the base has sidelength 2. Then the area of the base is the square of this, or 4.

The height of the pyramid is equal to the perimeter of the base, or $4 \times 2 = 8$ . A right triangle can be formed with the lengths of its legs equal to the height of the pyramid, or 8, and one half the length of a side, or 1; the length of its hypotenuse, which is the slant height, is

$\sqrt{8^{2} + 1 ^{2}} = \sqrt{64+ 1 } = \sqrt{65}$

This is the height of one triangular face; its base is a side of the square, so the length of the base is 2. The area of a face is half the product of these dimensions, or

$\frac{1}{2} \times 2 \times \sqrt{65} = \sqrt{65}$

Since twice the area of the base is $8 = \sqrt{64}$ , the problem comes down to comparing $\sqrt{64}$ and $\sqrt{65}$ ; the latter, which is (B), is greater.

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Example Question #1 : Pyramids

A pyramid with a square base and a cone have the same height and the same volume. Which is the greater quantity?

(A) The perimeter of the base of the pyramid

(B) The circumference of the base of the cone

Possible Answers:

It is impossible to determine which is greater from the information given

(A) is greater

(B) is greater

(A) and (B) are equal

Correct answer:

(A) is greater

Explanation:

The volume of a pyramid or a cone with height and base of area is

$V = \frac{1}{3} Bh$ ,

so in both cases, the area of the base is

$\frac{1}{3} Bh = V$

$\frac{3}{h} \cdot \frac{1}{3} Bh =\frac{3}{h} \cdot V$

$B = \frac{3V}{h}$

Since the pyramid and the cone have the same volume and height, their bases has the same area .

The length of one side of the square base of the pyramid is the square root of this, or $\sqrt{B}$ , and the perimeter is four times this, or $P = 4 \sqrt{B}$ .

The radius and the area of the base of the cone are related as follows:

$\pi r ^{2} = B$

$\sqrt{\pi r ^{2} }= \sqrt{B}$

$\sqrt{\pi } \cdot r = \sqrt{B}$

Multiply both sides by $2 \sqrt{\pi}$ to get:

$2 \sqrt{\pi} \cdot \sqrt{\pi } \cdot r =2 \sqrt{\pi} \cdot \sqrt{B}$

$2\pi r =2 \sqrt{\pi} \cdot \sqrt{B}$

$C =2 \sqrt{\pi} \cdot \sqrt{B}$

$2 \sqrt{\pi} < 2 \sqrt{4} = 2 \times 2 = 4$ , so

$2 \sqrt{\pi} \cdot \sqrt{B} < 4 \sqrt{B}$ , and

C < P

The perimeter of the base of the pyramid, which is (A), is greater than the circumference of the base of the cone.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Pyramids

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

All ISEE Upper Level Quantitative Resources

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 #1 : How To Find The Volume Of A Pyramid

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

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

Example Question #1 : Pyramids

All ISEE Upper Level Quantitative Resources

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