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ACT Math : Solid Geometry

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Example Questions

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

If a cylinder has a radius, $\small r$ , of 2 inches and a height, $\small h$ , of 5 inches, what is the total surface area of the cylinder?

Possible Answers:

$\small 28\pi$

$\small 36\pi$

$\small 24\pi$

$\small 70\pi$

$\small 18\pi$

Correct answer:

$\small 28\pi$

Explanation:

The total surface area will be equal to the area of the two bases added to the area of the outer surface of the cylinder. If "unwrapped" the area of the outer surface is simply a rectangle with the height of the cylinder and a base equal to the circumference of the cylinder base. We can use these relationships to find a formula for the total area of the cylinder.

$A=2A_{base}+A_{rectangle}$

$A=2(\pi r^2)+(2\pi r)(h)$

Use the given radius and height to solve for the final area.

$\small 2\pi(2)^{2} + 2\pi (2)(5)$

$\small 8\pi + 20\pi$

$\small 28\pi$

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Example Question #2 : How To Find The Surface Area Of A Cylinder

What is the surface area of a cylinder with a base diameter of 6in and a height of 8in ?

Possible Answers:

$36\pi in^{2}$

$66\pi in^{2}$

$48\pi in^{2}$

$14\pi in^{2}$

None of the answers

Correct answer:

$66\pi in^{2}$

Explanation:

Area of a circle $A=\pi r^{2}=\pi(\frac{d}{2})^{2}=\pi(\frac{6}{2})^{2}=\pi(3)^{2}=9\pi in^{2}$

Circumference of a circle $C=\pi d=6\pi in^{2}$

Surface area of a cylinder $SA=2A+hC=2(9\pi)+8(6\pi)=18\pi+48\pi=66\pi in^{2}$

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Example Question #15 : Cylinders

What is the surface area of a cylinder with a radius of and a height of ? Give your answer in terms of $\pi$ .

Possible Answers:

$120\pi$

$55\pi$

$95\pi$

$170\pi$

$168\pi$

Correct answer:

$120\pi$

Explanation:

To find the surface area of a cylinder with radius and height use the equation:

$SA = 2*\pi*r^2 + 2*\pi*r*h$
Thus for a cylinder with a radius of 5 and a height of 7 we get:

$120\pi$

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Example Question #1041 : Act Math

Find the surface area of a cylinder whose height is 6 and radius is 7.

Possible Answers:

$64\pi$

$56\pi$

$28\pi$

$182\pi$

Correct answer:

$182\pi$

Explanation:

To solve, simply use the formula for surface area of a cylinder.

First, identify all known information.

Height = 6

Radius = 7

Substitute these values into the surface area equation and solve.

Thus,

$SA=2\pi{r}h+2\pi{r^2}=2*\pi*7*6+2*\pi*7^2=84\pi+98\pi=182\pi$

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Example Question #16 : Cylinders

A grain silo in the shape of a right circular cylinder is erected vertically, as shown below. The silo is then covered with corrugated steel. If the cylinder is $12\ $m$$ tall and has a circumference of $6\pi \ $m$$ , how much corrugated steel, in square meters, must be used to cover the visible portion of the silo?

Possible Answers:

$100\pi$

$81\pi$

$64\pi$

$128\pi$

$107\pi$

Correct answer:

$81\pi$

Explanation:

The surface area of a cylinder can be calculated using the following formula:

$SA = 2\pi rh + 2\pi r^2$

In this equation, the variable, , is the radius of the base of the cylinder and is the height of the cylinder.

In this case, we must also remember not to include one of the two $\pi r^2$ measurements, since the bottom face that is in contact with the ground will not be covered with corrugated steel. We need to modify our surface area formula in the following way:

$SA = 2\pi rh + \pi r^2$

We are given the circumference of the cylinder; therefore, we can use this information to solve for the radius.

$2\pi r=6\pi$

r=3

Since the circumference is $6\pi$ , we know the radius is 3. Now we can insert these values to the modified surface area equation and solve for the coverable surface area of the silo.

$SA = 2\pi rh + \pi r^2$

$SA = 6\pi(12) + \pi(3^2)$

$SA = 72\pi+ 9\pi$

$SA = 81\pi \textup{ m}^2$

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Example Question #21 : Cylinders

The volume of a cylinder is $81\pi$ . If the radius of the cylinder is , what is the surface area of the cylinder?

Possible Answers:

$3\pi$

$72\pi$

$27\pi$

$9\pi$

$81\pi$

Correct answer:

$72\pi$

Explanation:

The volume of a cylinder is equal to:

$V=\pi r^2h$

Use this formula and the given radius to solve for the height.

$81\pi=\pi(3)^2h$

$81\pi=9\pi(h)$

h=9

Now that we know the height, we can solve for the surface area. The surface area of a cylinder is equal to the area of the two bases plus the area of the outer surface. The outer surface can be "unwrapped" to form a rectangle with a height equal to the cylinder height and a base equal to the circumference of the cylinder base. Add the areas of the two bases and this rectangle to find the total area.

$A=2A_{base}+A_{rec}$

$A=2\pi r^2+(2\pi r)(h)$

Use the radius and height to solve.

$A=2\pi (3)^2+2\pi (3)(9)$

$A=18\pi+54\pi$

$A=72\pi$

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Example Question #12 : Cones

What is the surface area of a cone with a radius of 6 in and a height of 8 in?

Possible Answers:

36π in²

60π in²

66π in²

112π in²

96π in²

Correct answer:

96π in²

Explanation:

Find the slant height of the cone using the Pythagorean theorem: r² + h² = s² resulting in 6² + 8² = s² leading to s² = 100 or s = 10 in

SA = πrs + πr² = π(6)(10) + π(6)² = 60π + 36π = 96π in²

60π in² is the area of the cone without the base.

36π in² is the area of the base only.

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

Use the following formula to answer the question.

$SA=\pi r^2+\pi rl$

The slant height of a right circular cone is 6cm . The radius is 2cm , and the height is 4cm . Determine the surface area of the cone.

Possible Answers:

$12\pi cm^2$

$16\pi cm^2$

$18\pi cm^2$

$24\pi cm^2$

$20\pi cm^2$

Correct answer:

$16\pi cm^2$

Explanation:

Notice that the height of the cone is not needed to answer this question and is simply extraneous information. We are told that the radius is 2cm , and the slant height is 6cm .

First plug these numbers into the equation provided.

$SA=\pi r^2+\pi rl=\pi (2)^2+\pi (2)(6)=4\pi cm^2+12\pi cm^2$

Then simplify by combining like terms.

$4\pi cm^2+12\pi cm^2=16\pi cm^2$

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Example Question #11 : Surface Area

The slant height of a cone is ; the diameter of its base is one-fifth its slant height. Give the surface area of the cone in terms of .

Possible Answers:

$\frac{ \pi L^{2}}{10}$

$\frac{ \pi L^{2}}{20}$

$\frac{ \pi L^{2}}{5}$

$\frac{6\pi L^{2}}{25}$

$\frac{11\pi L^{2}}{100}$

Correct answer:

$\frac{11\pi L^{2}}{100}$

Explanation:

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r L$ .

The diameter of the base is $\frac{1}{5}L$ ; the radius is half this, so

$r = \frac{1}{2} \cdot \frac{1}{5}L= \frac{1}{10}L$

Substitute in the surface area formula:

$A = \pi \left ( \frac{1}{10}L \right ) ^{2} + \pi \left ( \frac{1}{10}L \right ) L$

$A = \frac{1}{100} \pi L^{2} + \frac{1}{10} \pi L^{2}$

$A = \frac{1}{100} \pi L^{2} + \frac{10}{100} \pi L^{2}$

$A = \frac{11}{100} \pi L^{2} = \frac{11\pi L^{2}}{100}$

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

The radius of the base of a cone is ; its slant height is two-thirds of the diameter of that base. Give its surface area in terms of .

Possible Answers:

$\frac{5\pi r ^{2}}{3}$

$\frac{4\pi r ^{2}}{3}$

$\frac{7\pi r ^{2}}{3}$

$\frac{\left (6- \sqrt{7} \right )\pi r ^{2}}{3}$

$\frac{\left (3- \sqrt{5} \right )\pi r ^{2}}{3}$

Correct answer:

$\frac{7\pi r ^{2}}{3}$

Explanation:

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r l$ .

The diameter of the base is twice radius , or , and its slant height is two-thirds of this diameter, which is $\frac{2}{3}\cdot 2r = \frac{4}{3}r$ . Substitute this for in the formula:

$A = \pi r^{2} + \pi r \cdot \frac{4}{3}r$

$A = \pi r^{2} + \frac{4}{3} \pi r ^{2}$

$A = \frac{7}{3} \pi r ^{2} = \frac{7\pi r ^{2}}{3}$

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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 Surface Area Of A Cylinder

Example Question #2 : How To Find The Surface Area Of A Cylinder

Example Question #15 : Cylinders

Example Question #1041 : Act Math

Example Question #16 : Cylinders

Example Question #21 : Cylinders

Example Question #12 : Cones

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

Example Question #11 : Surface Area

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

All ACT Math Resources

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