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ISEE Upper Level Math : Solid Geometry

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Example Questions

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

Find the surface area of a non-cubic prism with the following measurements:

l=7;w=6;h=2

Possible Answers:

136

168

Correct answer:

136

Explanation:

The surface area of a non-cubic prism can be determined using the equation:

SA=2lw+2wh+2lh

SA=2(7)(6)+2(6)(2)+2(7)(2)=84+24+28=136

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Example Question #2 : Solid Geometry

Box

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the surface area of the box.

Possible Answers:

60x + 450

$4x^{2} + 30 x$

$15x^{2}$

225x

Correct answer:

60x + 450

Explanation:

A square has four sides of equal length, as seen in the diagram below.

Box

All six sides are rectangles, so their areas are equal to the products of their dimensions:

Top, bottom, front, back (four surfaces): 15x

Left, right (two surfaces): $15 \cdot 15 =225$

The total surface area: $4 \cdot 15x + 2 \cdot 225 = 60x + 450$

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Example Question #1 : Finding Volume Of A Rectangular Prism

A rectangular prism has a width of 3 inches, a length of 6 inches, and a height triple its length. Find the volume of the prism.

Possible Answers:

38in^3

420in^3

27in^3

324in^3

Correct answer:

324in^3

Explanation:

A rectangular prism has a width of 3 inches, a length of 6 inches, and a height triple its length. Find the volume of the prism.

Find the volume of a rectangular prism via the following:

V=l*w*h

Where l, w, and h are the length width and height, respectively.

We know our length and width, and we are told that our height is triple the length, so...

h=3l=3*6in=18in

Now that we have all our measurements, plug them in and solve:

V=3in*6in*18in=324in^3

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Example Question #1 : Finding Volume Of A Rectangular Prism

Box

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the volume of the box.

Possible Answers:

225x

$4x^{2} + 30 x$

$15x^{2}$

60x + 450

Correct answer:

225x

Explanation:

A square has four sides of equal length, as seen in the diagram below.

Box

The volume of the solid is equal to the product of its length, width, and height, as follows:

$V = 15 \cdot 15 \cdot x =225 x$ .

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Example Question #1 : Pyramids

A pyramid has height 4 feet. Its base is a square with sidelength 3 feet. Give its volume in cubic inches.

Possible Answers:

$5,184\textrm{ in}^{3}$

$124,416\textrm{ in}^{3}$

$62,208 \textrm{ in}^{3}$

$31,104\textrm{ in}^{3}$

$20,736 \textrm{ in}^{3}$

Correct answer:

$20,736 \textrm{ in}^{3}$

Explanation:

Convert each measurement from inches to feet by multiplying it by 12:

Height: 4 feet = $4 \times 12 = 48$ inches

Sidelength of the base: 3 feet = $3 \times 12 = 36$ inches

The volume of a pyramid is

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

Since the base is a square, we can replace $B = s^{2}$ :

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

Substitute s=36, h = 48

$V = \frac{1}{3} \cdot 36 ^{2}\cdot 48$

V =20,736

The pyramid has volume 20,736 cubic inches.

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

A 300 foot tall pyramid has a square base measuring feet on each side. What is the volume of the pyramid?

Possible Answers:

$160,000 ft^{3}$

$12,000 ft^{3}$

$480,000 ft^{3}$

$4,000 ft^{3}$

$160,000 ft^{2}$

Correct answer:

$160,000 ft^{3}$

Explanation:

In order to find the area of a triangle, we use the formula $\frac{1}{3}Bh$ . In this case, since the base is a square, we can replace with $s^{2}$ , so our formula for volume is $\frac{1}{3}s^{2}h$ .

Since the length of each side of the base is feet, we can substitute it in for .

$\frac{1}{3}\cdot (40)^{2}\cdot h$

We also know that the height is 300 feet, so we can substitute that in for .

$\frac{1}{3}\cdot (40)^{2}\cdot 300$

This gives us an answer of $160,000 ft^{3}$ .

It is important to remember that volume is expressed in units cubed.

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Example Question #3 : Pyramids

The height of a right pyramid is feet. Its base is a square with sidelength feet. Give its volume in cubic inches.

Possible Answers:

$15,552 \textrm{ in}^{3}$

$41,472 \textrm{ in}^{3}$

$10,368 \textrm{ in}^{3}$

$31,104 \textrm{ in}^{3}$

$62,208\textrm{ in}^{3}$

Correct answer:

$10,368 \textrm{ in}^{3}$

Explanation:

Convert each of the measurements from feet to inches by multiplying by .

Height: $2 \times 12 = 24$ inches

Sidelength of base: $3 \times 12 = 36$ inches

The base of the pyramid has area

$B = s^{2} = 36^{2} = 1,296$ square inches.

Substitute B = 1,296, h = 24 into the volume formula:

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

$V = \frac {1}{3} \cdot 1,296 \cdot 24 =10,368$ cubic inches

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

The height of a right pyramid is inches. Its base is a square with sidelength inches. Give its volume in cubic feet.

Possible Answers:

$9 \frac {1}{3} \textrm{ ft}^3$

$4 \frac {2}{3} \textrm{ ft}^3$

$2 \frac {1}{3} \textrm{ ft}^3$

$14 \textrm{ ft}^3$

$4 \textrm{ ft}^3$

Correct answer:

$4 \frac {2}{3} \textrm{ ft}^3$

Explanation:

Convert each of the measurements from inches to feet by dividing by .

Height: $42 \div 12 = 3 \frac{1}{2}$ feet

Sidelength: $24 \div 12 = 2$ feet

The base of the pyramid has area

$B = s^{2} = 2^{2} = 4$ square feet.

Substitute $B = 4, h = 3 \frac{1}{2}$ into the volume formula:

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

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

$V = \frac {1}{3} \cdot \frac {4}{1} \cdot \frac{7}{2}$

$V = \frac {1}{3} \cdot \frac {2}{1} \cdot \frac{7}{1} = \frac {14}{3} = 4 \frac {2}{3}$ cubic feet

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Example Question #3 : Volume Of A Pyramid

The height of a right pyramid and the sidelength of its square base are equal. The perimeter of the base is 3 feet. Give its volume in cubic inches.

Possible Answers:

$243 \textrm{ in}^{3}$

$27 \textrm{ in}^{3}$

$576 \textrm{ in}^{3}$

$81 \textrm{ in}^{3}$

$729 \textrm{ in}^{3}$

Correct answer:

$243 \textrm{ in}^{3}$

Explanation:

The perimeter of the square base, feet, is equivalent to $3 \times 12 = 36$ inches; divide by to get the sidelength of the base - and the height: $36 \div 4 = 9$ inches.

The area of the base is therefore $B = s^{2} = 9^{2} = 81$ square inches.

In the formula for the volume of a pyramid, substitute B = 81,h=9 :

$V = \frac{1}{3} Bh = \frac{1}{3}\cdot 81 \cdot 9 = 243$ cubic inches.

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Example Question #2 : Pyramids

What is the volume of a pyramid with the following measurements?

$length=7; width=6;height=9;slant\ length=11$

Possible Answers:

154

126

462

378

Correct answer:

126

Explanation:

The volume of a pyramid can be determined using the following equation:

$V=\frac{1}{3}lwh=\frac{1}{3}(7)(6)(9)=126$

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Solid Geometry

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

All ISEE Upper Level Math Resources

Example Questions

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

Example Question #2 : Solid Geometry

Example Question #1 : Finding Volume Of A Rectangular Prism

Example Question #1 : Finding Volume Of A Rectangular Prism

Example Question #1 : Pyramids

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

Example Question #3 : Pyramids

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

Example Question #3 : Volume Of A Pyramid

Example Question #2 : Pyramids

All ISEE Upper Level Math Resources

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