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SAT Math : Pyramids

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Example Questions

Example Question #1 : Pyramids

If the height of a pyramid was increased by 20% and a side of the square base was decreased 30%, what would happen to the volume of the pyramid?

Possible Answers:

There is a 59% increase in volume

It would have the same volume

59% decrease in volume

There is a 41% decrease in volume

There is no way to know if it would increase or decrease in volume

Correct answer:

There is a 41% decrease in volume

Explanation:

First, you will want to create a pyramid with measurements that are easy to calculate. So, let's say that we have pyramid with a base edge of 10 inches and a height of 10 inches.

So the volume of the original pyramid would be equal to

$\frac{(10\times 10)(10)}{3}=\frac{1000}{3}=333.33\: inches^2$

The volume of the altered pyramid would be equal to:

$\frac{(7\times 7)(12)}{3}=\frac{588}{3}=196\: inches^2$

To find the relationship between the volume of the altered pyramid relative to the volume of the original pyramid, divide the altered volume by the original volume.

$\frac{196}{333.33}=.59$

The new volume is 59% of the original volume, which means there was a 41% decrease in volume.

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Example Question #1 : Pyramids

The Pyramid of Giza has a height of 480 feet. If the length of each side of the base is approximately 756 feet, what is its total surface area? Round to the nearest tenth.

Possible Answers:

$934,416 ft^{2}$

$802,494 ft^{2}$

$1,495,368 ft^{2}$

$752,976 ft^{2}$

Correct answer:

$1,495,368 ft^{2}$

Explanation:

If the length of one side is 756 ft, then multiply to find the area of the base.

$\text{Area of square} = \text{length} \times \text{width}$

$A=756\times756$

$A=571,536 ft^{2}$

Once you've found the area of the base, use the height of the pyramid and half of the side length of the base to determine the length of the side from the apex to the ground using the Pythagorean Theorem.

$480^{2}+378^{2}=c^{2}$

$230,400+142,884=c^{2}$

$373,284=c^{2}$

611=c

Using the side length of the base and the height of each of the triangles that form the pyramid, calculate the area of each triangle, then multiply by 4.

$\text{Area of a triangle}=\frac{1}{2}\times \text{base} \times \text{height}$

$A=\frac{1}{2} \times 611 \times 756$

A=230,958

$\text{Area of four triangles} = 4 \times 230,958$

$A=923,832 ft^{2}$

Add the surface area of the base to the surface area of the four triangles.

$\text{Total surface area} = 571,536 + 923,832$

$1,495,368 ft^{2}$

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Example Question #2 : Pyramids

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

Possible Answers:

$2\ ft$

$4\ ft$

$4\ ft^{2}$

$1.5\ ft$

$2\ ft^{2}$

Correct answer:

$2\ ft$

Explanation:

Volume of a pyramid is

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

Thus:

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

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

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

Therefore, each side is $2\ ft$ .

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Example Question #1 : 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 #52 : Solid Geometry

Find the volume of the pyramid shown below:

Screen shot 2015 10 27 at 3.33.57 pm

Possible Answers:

689

432

$\frac{17}{3}$

144

Correct answer:

144

Explanation:

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

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

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Example Question #4 : Pyramids

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:

$12\ cm^3$

$25\ cm^3$

$24\ cm^3$

$10\ cm^3$

$30\ cm^3$

Correct answer:

$24\ cm^3$

Explanation:

Pyramid question notes1

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

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

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.

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

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

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

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

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

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

Now we plug in these values to find the volume:

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

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

$Volume=24\, in^3$

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Example Question #3 : Pyramids

Pyramid

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

Possible Answers:

72in.^3

32in.^3

60in.^3

48in.^3

36in.^3

Correct answer:

48in.^3

Explanation:

The volume of a rectangular pyramid with height , base width , and base length is given by

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

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

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

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

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SAT Math : Pyramids

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : Pyramids

Example Question #1 : Pyramids

Example Question #2 : Pyramids

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

Example Question #52 : Solid Geometry

Example Question #4 : Pyramids

Example Question #3 : Pyramids

All SAT Math Resources

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