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ACT Math : Solid Geometry

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Example Questions

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

A pyramid is placed inside a cube so that they share a base and height. If the surface area of the cube is $486\textup{ft}^{2}$ , what is the volume of the pyramid, in square feet?

Possible Answers:

$216\textup{ft}^{2}$

$243\textup{ft}^{2}$

$729\textup{ft}^{2}$

$225\textup{ft}^{2}$

Correct answer:

$243\textup{ft}^{2}$

Explanation:

First, we need to find the length of a side for the cube.

Recall that the surface area of the cube is given by the following equation:

$\text{Surface Area}=6s^2$ , where is the length of a side.

Plugging in the surface area given by the equation, we can then find the side length of the cube.

486=6s^2

s^2=81

s=9

Now, because the pyramid and the cube share a base, we know that the pyramid must be a square pryamid.

Recall how to find the volume of a pyramid:

$\text{Volume}=\text{Area of base}\times \text{height}\times \frac{1}{3}$

$\text{Area of Base}=9\times9=81$

Now, since the pyramid is the same height as the cube, the height of the pyramid is also .

$\text{Volume}=81\times 9\times \frac{1}{3}=243$

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Example Question #11 : How To Find The Volume Of A Polyhedron

In cubic feet, find the volume of the pentagonal prism illustrated below. The pentagon has an area of $16\textup{ft}^{2}$ , and the prism has a height of $8\textup{ft}$ .

Possible Answers:

$224\textup{ft}^{3}$

$20\textup{ft}^{3}$

$128\textup{ft}^{3}$

$96\textup{ft}^{3}$

Correct answer:

$128\textup{ft}^{3}$

Explanation:

For any prism, the volume is given by the following equation:

$\text{Volume}=\text{Area of base}\times\text{height}$

The question gives us the area of the base and the height.

$\text{Volume}=16\times8=128$

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Example Question #641 : Geometry

If the side lengths of a cube are tripled, what effect will it have on the volume?

Possible Answers:

The volume will be times as large.

Correct answer:

The volume will be times as large.

Explanation:

Start by taking a cube that is $1\times1\times1$ . The volume of this cube is .

Next, triple the sides of this cube so that it becomes $3\times3\times3$ . The volume of this cube is .

The volume of the new cube is times as large as the original.

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Example Question #641 : Geometry

The surface area of a cylinder is given by $2\pi rh+2\pi r^{2}$ , where is the radius and is the height. If a cylinder has a surface area of $20\pi$ $cm^{2}$ and a height of , what is its radius?

Possible Answers:

1cm

5 cm

2 cm

Correct answer:

2 cm

Explanation:

The fastest way to solve this problem is to plug all of the answer choices in for and look for an output of $20\pi$ .

Alternatively, one could set the surface area formula equal to $20\pi$ (knowing that h=3cm ), but this would require solving a quadratic.

$20\pi =2\pi r(3)+2\pi r^2$

$0=6\pi r +2\pi r^2-20\pi$

Rearranging this we get

$2\pi r^2+6 \pi r-20\pi=0$

Now factoring out a $2\pi$ we get:

$2\pi (r^2+3r-10)=0$

$2\pi(r+5)(r-2)=0$

$r+5=0 \rightarrow r=-5$

$r-2=0 \rightarrow r=2$

Since we cannot have a negative length our answer is 2 cm .

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

If a half cylinder with a height of 5 and semicircular bases with a radius of 2, what is the surface area?

Possible Answers:

$20+10\pi$

$20+14\pi$

$20\pi+14$

$34\pi$

$20\pi+10$

Correct answer:

$20+14\pi$

Explanation:

The surface area of the half cylinder will consist of the lateral area and the base area.

There are three parts to the surface area:

Rectangular region, semi-circular bases of the half cylinder, and outer face of the cylinder.

The cross section of the half cylinder is a rectangle. Find the area of the rectangle. The diameter of the semicircle represents the width of the rectangle, which is double the radius. The length of the rectangle is the height of the cylinder.

$A_{\textup{rectangle}}=LW=(4)(5)=20$

Next, find the area of a semicircular bases.

$A_{\textup{semicircular base}} = \frac{\pi r^2}{2} = \frac{\pi (2^2)}{2}= 2\pi$

Since there are two semicircular bases of the semi-cylinder, the total area of the semicircular bases is $4\pi$ .

$A_{\textup{ TWO semicircular bases}} = 4\pi$

Find the area of the outer region. The area of the outer region is the half circumference multiplied by the height of the cylinder.

$A_{\textup{outer region}}= \frac{2\pi r}{2} \times h = 2\pi (5)= 10\pi$

Sum the areas of the rectangle, the two circular bases, and the outer region to find the lateral area.

$\sum A=20 + 4\pi + 10\pi = 20+14\pi$

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Example Question #1 : How To Find The Length Of An Edge Of A Cube

Our backyard pool holds 10,000 gallons. Its average depth is 4 feet deep and it is 10 feet long. If there are 7.48 gallons in a cubic foot, how wide is the pool?

Possible Answers:

33 ft

133 ft

30 ft

100 ft

7.48 ft

Correct answer:

33 ft

Explanation:

There are 7.48 gallons in cubic foot. Set up a ratio:

1 ft³ / 7.48 gallons = x cubic feet / 10,000 gallons

Pool Volume = 10,000 gallons = 10,000 gallons * (1 ft³/ 7.48 gallons) = 1336.9 ft³

Pool Volume = 4ft x 10 ft x WIDTH = 1336.9 cubic feet

Solve for WIDTH:

4 ft x 10 ft x WIDTH = 1336.9 cubic feet

WIDTH = 1336.9 / (4 x 10) = 33.4 ft

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Example Question #2 : How To Find The Length Of An Edge Of A Cube

A cube has a volume of 64cm³. What is the area of one side of the cube?

Possible Answers:

16cm²

4cm

4cm²

16cm³

16cm

Correct answer:

16cm²

Explanation:

The cube has a volume of 64cm³, making the length of one edge 4cm (4 * 4 * 4 = 64).

So the area of one side is 4 * 4 = 16cm²

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Example Question #1 : How To Find The Length Of An Edge Of A Cube

A cube as the volume of 64 units^3 .

Find the length of a side of this cube.

Possible Answers:

Correct answer:

Explanation:

The formula to find the volume of the cube is

(side)^3

Since we know the volume, we can set up the equation

(side)^3=64

$\sqrt[3]64=4$

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Example Question #11 : Solid Geometry

A cube has a surface area of $1014\textup{ in}^2$ , what is the length of the side of the cube? (If necessary, round to the nearest hundredth.)

Possible Answers:

$29.12\textup{ inches}$

$23.47\textup{ inches}$

$169\textup{ inches}$

$13\textup{ inches}$

$31.84\textup{ inches}$

Correct answer:

$13\textup{ inches}$

Explanation:

To find the length of the side of a square given the surface area, use the surface area formula and solve for :

$1014\textup{ inches}^2 = 6*s^2$ , now divide both sides by 6
$169\textup{ inches}^2 = s^2$ , now square root both sides
13 = s .

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Example Question #2 : How To Find The Length Of An Edge Of A Cube

A certain cubic box when unfolded and laid flat on a table covers exactly 360 square units of space. What is the width of the box, in units?

Possible Answers:

$6\sqrt{6}$

$2\sqrt{15}$

Correct answer:

$2\sqrt{15}$

Explanation:

To find the length of the edge of a cube from its surface area, remember that $SA\square = 6s^2$ , where is the length of a side.

6s^2 = 360

$s = \sqrt{60} = 2\sqrt{15}$

So, the box is $2\sqrt{15}$ units long.

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ACT Math : Solid Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

Example Question #11 : How To Find The Volume Of A Polyhedron

Example Question #641 : Geometry

Example Question #641 : Geometry

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

Example Question #1 : How To Find The Length Of An Edge Of A Cube

Example Question #2 : How To Find The Length Of An Edge Of A Cube

Example Question #1 : How To Find The Length Of An Edge Of A Cube

Example Question #11 : Solid Geometry

Example Question #2 : How To Find The Length Of An Edge Of A Cube

All ACT Math Resources

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