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ACT Math : Geometry

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Example Questions

Example Question #761 : Geometry

Find the volume of a cylinder given height of and radius of .

Possible Answers:

$32\pi$

$128\pi$

$16\pi$

$64\pi$

Correct answer:

$32\pi$

Explanation:

To solve, simply use the following formula. Thus,

$V=\pi{r^2}h=\pi*2^2*8=\pi*4*8=32\pi$

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Example Question #13 : Cylinders

A cylindrical tank is used as part of a water purifying plant. When contaminated water flows into the top section of the tank, pressure forces it through a mesh filter at the bottom of the tank and clean water exits through a funnel, leaving sediment behind. The tank's filter must be replaced when the total sediment content of the tank exceeds ten percent of the tank's total volume. If the tank is 100 feet tall and 18 feet in diameter, how much sediment, in cubic feet, can the drum hold before the filter must be changed?

Possible Answers:

$900\pi$

$760\pi$

$810\pi$

$8100\pi$

$9000\pi$

Correct answer:

$810\pi$

Explanation:

The volume of a cylinder is found using the following formula:

$V = \pi r^2 h$

In this formula, the variable is the height of the cylinder and is its radius. Since the diameter is two times the radius, first solve for the radius.

$\text{Diameter}=2r$

18=2r

Divide both sides of the equation by 2.

$\frac{18}{2}=\frac{2r}{}2$

r=9

The given cylinder has a radius of 9 feet. Now, substitute the calculated and known values into the equation for the volume of a cylinder and solve.

$V=\pi\times9^{2}\times 100$

$V=8100\pi$

This is the total volume of the tank. The question asks for the volume of ten percent of the tank—the point at which the filter must be replaced. To find this, move the decimal point in the numerical measure of total volume to the left one place in order to calculate ten percent of the total volume. (Ignore $\pi$ —you can treat it like a multiplier here. Since it appears on both sides of the equals sign, it doesn't affect the decimal shift.)

$8100\pi\times 0.10=810\pi$

The tank can hold $810\pi$ cubic feet of sediment before the filter needs to be changed.

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

What is the surface area of a cylinder with a radius of 2 cm and a height of 10 cm?

Possible Answers:

48π cm²

40π cm²

56π cm²

36π cm²

32π cm²

Correct answer:

48π cm²

Explanation:

SA_cylinder = 2πrh + 2πr² = 2π(2)(10) + 2π(2)² = 40π + 8π = 48π cm²

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Example Question #2 : 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:

$\small 36\pi$

$\small 28\pi$

$\small 18\pi$

$\small 24\pi$

$\small 70\pi$

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 #3 : 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:

$14\pi in^{2}$

$36\pi in^{2}$

$48\pi in^{2}$

$66\pi in^{2}$

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 #4 : How To Find The Surface Area Of A Cylinder

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:

$3\pi$

$81\pi$

$9\pi$

$27\pi$

$72\pi$

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 #42 : 3 Dimensional Geometry

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

Possible Answers:

96π in²

60π in²

36π in²

112π in²

66π 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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ACT Math : Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #761 : Geometry

Example Question #13 : Cylinders

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 #3 : How To Find The Surface Area Of A Cylinder

Example Question #15 : Cylinders

Example Question #1041 : Act Math

Example Question #16 : Cylinders

Example Question #4 : How To Find The Surface Area Of A Cylinder

Example Question #42 : 3 Dimensional Geometry

All ACT Math Resources

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