Surface Area and Volume – ASVAB Math Review

1.Cube

Total surface and space inside a cube

$$V = s^3$$ SA $$= 6 \times s^2$$

s: side length, V: Volume, SA: Surface Area.

Example

A wooden cube has a side length of 4 inches. Find its total surface area and volume.

Show solution

Step 1: Calculate Surface Area by squaring the side and multiplying by 6: $$6 \times (4^2) = 6 \times 16 = 96$$ square inches

Step 2: Calculate Volume by cubing the side length: $$4^3 = 4 \times 4 \times 4 = 64$$ cubic inches

Practice problems

Problem 1
A metal shipping container is shaped like a cube with a side length of 5 meters. What is the total volume of the container?

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Answer: 125 cubic meters

Step 1: Identify the side length: $$s = 5$$ meters
Step 2: Apply the volume formula for a cube: $$V = s^3 = 5 \times 5 \times 5$$
Step 3: Calculate the final value: $$V = 125$$ cubic meters
Problem 2
Find the total surface area of a cardboard box shaped like a cube with an edge measuring 3 feet.

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Answer: 54 square feet

Step 1: Identify the side length parameter: $$s = 3$$ feet
Step 2: Apply the surface area formula: SA $$= 6 \times s^2 = 6 \times (3 \times 3)$$
Step 3: Complete the multiplication: $$6 \times 9 = 54$$ square feet
Problem 3
If the volume of a cubic die is 8 cubic centimeters, what is the length of one of its edges?

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Answer: 2 cm

Step 1: Set up the volume equation: $$8 = s^3$$
Step 2: Find the cube root of 8 to isolate s: $$s = 2$$
Step 3: Confirm the edge length: 2 cm

Remember

For Surface Area, square the side length first, then multiply by 6. For Volume, cube the side length $$(s \times s \times s)$$

2.Rectangular Prism

Box-shaped object

$$V = l \times w \times h$$, SA = 2(lw + lh + wh)

l: length, w: width, h: height.

Example

A rectangular brick has a length of 8 inches, a width of 4 inches, and a height of 3 inches. Find its volume.

Show solution

Step 1: Identify the given dimensions: $$l = 8$$, $$w = 4$$, $$h = 3$$

Step 2: Apply the volume formula: $$V = 8 \times 4 \times 3$$

Step 3: Compute the product value: $$32 \times 3 = 96$$ cubic inches

Practice problems

Problem 4
A storage room measures 10 feet long, 6 feet wide, and 8 feet high. What is the total volume space of the room?

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Answer: 480 cubic feet

Step 1: Assign the dimensions: $$l = 10$$, $$w = 6$$, $$h = 8$$
Step 2: Use the volume formula: $$V = 10 \times 6 \times 8$$
Step 3: Multiply out the values: $$60 \times 8 = 480$$ cubic feet
Problem 5
Calculate the total surface area of a box-shaped wooden block that is 5 cm long, 3 cm wide, and 2 cm high.

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Answer: 62 square cm

Step 1: Identify dimensions: $$l = 5$$, $$w = 3$$, $$h = 2$$
Step 2: Multiply the paired combinations: lw = 15, lh = 10, wh = 6
Step 3: Add the pairs and multiply by 2: SA $$= 2 \times (15 + 10 + 6) = 2 \times 31 = 62$$ square cm
Problem 6
A rectangular water tank holds a volume of 120 cubic meters. If the base length is 6 meters and the width is 4 meters, find the height of the tank.

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Answer: 5 meters

Step 1: Set up the volume formula: $$120 = 6 \times 4 \times h$$-> $$120 = 24 \times h$$
Step 2: Isolate the height variable: $$h = \frac{120}{24}$$
Step 3: Divide to find the final height: $$h = 5$$ meters

Remember

When calculating surface area, compute the three separate paired multiplication sets inside the parentheses before multiplying by 2.

3.Cylinder

Soda can shape

$$V = \pi \times r^2 \times h$$, SA $$= 2\pi r(r + h)$$

r: radius, h: height.

Example

A soda can has a base radius of 3 cm and a height of 10 cm. Find its volume.

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Step 1: Square the radius parameter: $$3 \times 3 = 9$$

Step 2: Multiply the squared radius by the height: $$9 \times 10 = 90$$

Step 3: Multiply by pi to get the final volume: $$V = 90 \times 3.14 = 282.6$$ cubic cm

Practice problems

Problem 7
A cylindrical water pipe has a base radius of 2 inches and a length height of 20 inches. What is the total volume of water it holds?

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Answer: 251.2 cubic inches

Step 1: Apply the volume formula parameters: $$r = 2$$, $$h = 20$$
Step 2: Calculate the base area times height: $$V = 3.14 \times (2^2) \times 20 = 3.14 \times 4 \times 20$$
Step 3: Complete the calculation: $$3.14 \times 80 = 251.2$$ cubic inches
Problem 8
Find the volume of a large oil drum shaped like a soda can that has a base diameter of 4 meters and a height of 5 meters.

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Answer: 62.8 cubic meters

Step 1: Convert the diameter to base radius: $$r = \frac{4}{2} = 2$$ meters
Step 2: Set up the volume equation: $$V = 3.14 \times (2^2) \times 5 = 3.14 \times 4 \times 5$$
Step 3: Calculate the product: $$3.14 \times 20 = 62.8$$ cubic meters
Problem 9
Find the total surface area of a cylinder container with a radius of 5 cm and a height of 10 cm.

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Answer: 471 square cm

Step 1: Identify variables: $$r = 5$$, $$h = 10$$
Step 2: Add radius and height: $$r + h = 5 + 10 = 15$$
Step 3: Multiply out the total formula: SA $$= 2 \times 3.14 \times 5 \times 15 = 31.4 \times 15 = 471$$ square cm

Remember

If the exam question gives you the total base diameter instead of the radius, always divide it by 2 first.

4.Sphere

Ball shape

$$V = (\frac{4}{3}) \times \pi \times r^3$$, SA $$= 4 \times \pi \times r^2$$

r: radius

Example

A rubber ball has a radius of 3 cm. Find its total surface area.

Show solution

Step 1: Square the radius parameter value: $$3^2 = 3 \times 3 = 9$$

Step 2: Set up the surface area equation: SA $$= 4 \times 3.14 \times 9$$

Step 3: Multiply values to get the surface area: SA $$= 36 \times 3.14 = 113.04$$ square cm

Practice problems

Problem 10
A round globe model is a perfect ball shape with a radius of 6 inches. Calculate its total outer surface area.

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Answer: 452.16 square inches

Step 1: Identify radius value: $$r = 6$$
Step 2: Apply the surface area formula: SA $$= 4 \times 3.14 \times (6^2) = 4 \times 3.14 \times 36$$
Step 3: Calculate the product value: $$144 \times 3.14 = 452.16$$ square inches
Problem 11
Find the internal volume of a ball-shaped container that has a core radius of 3 meters.

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Answer: 113.04 cubic meters

Step 1: Calculate the cubed radius value: $$r^3 = 3 \times 3 \times 3 = 27$$
Step 2: Apply the volume formula: $$V = (\frac{4}{3}) \times 3.14 \times 27$$
Step 3: Simplify the fraction multiplier: $$V = 4 \times 3.14 \times 9 = 36 \times 3.14 = 113.04$$ cubic meters
Problem 12
If a basketball has a total diameter of 20 cm, what is the correct radius parameter value to use in its volume formula?

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Answer: 10 cm

Step 1: Recognize that the provided value is the complete diameter (20 cm).
Step 2: Recall the radius rule formula: $$r = \text{diameter} / 2$$
Step 3: Divide to find the radius parameter value: $$\frac{20}{2} = 10$$ cm

Remember

For volume calculations, make sure to cube the radius $$(r \times r \times r)$$ before multiplying by $$\frac{4}{3}$$ and pi.

5.Cone

Ice cream cone shape

$$V = (\frac{1}{3}) \times \pi \times r^2 \times h$$, SA $$= \pi \times r(r + s)$$

r: radius, h: height, s: slant height.

Example

An ice cream cone shape has a base radius of 3 cm and a vertical height of 7 cm. Find its volume.

Show solution

Step 1: Square the radius parameter: $$3 \times 3 = 9$$

Step 2: Set up the cone volume formula: $$V = (\frac{1}{3}) \times (\frac{22}{7}) \times 9 \times 7$$

Step 3: Cross-simplify the values to solve: $$V = 3 \times 22 = 66$$ cubic cm

Practice problems

Problem 13
A conical pile of sand has a base radius of 6 feet and a vertical height of 10 feet. What is its volume?

Show solution

Answer: 376.8 cubic feet

Step 1: Square the radius: $$6^2 = 36$$
Step 2: Apply the volume formula: $$V = (\frac{1}{3}) \times 3.14 \times 36 \times 10$$
Step 3: Simplify the fraction and multiply: $$12 \times 3.14 \times 10 = 120 \times 3.14 = 376.8$$ cubic feet
Problem 14
Find the total surface area of a conical party hat that has a base radius of 3 inches and a slant height of 5 inches.

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Answer: 75.36 square inches

Step 1: Identify the given parameters: $$r = 3$$, $$s = 5$$
Step 2: Add the radius and slant height: $$r + s = 3 + 5 = 8$$
Step 3: Apply the surface area formula: SA $$= 3.14 \times 3 \times 8 = 9.42 \times 8 = 75.36$$ square inches
Problem 15
A cylinder and a cone share the exact same height and base radius. If the volume of the cylinder is 90 cubic meters, what is the volume of the cone?

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Answer: 30 cubic meters

Step 1: Recall that a cone's volume is exactly $$\frac{1}{3}$$ of a matching cylinder's volume.
Step 2: Set up the division calculation: $$\frac{90}{3}$$
Step 3: Evaluate the expression: 30 cubic meters

Remember

Make sure to differentiate between the vertical height (h) used for volume and the slant height (s) used for surface area.