4.2 Volume (rectangular prisms, cylinders)

Filling Tanks, Boxes, and Slabs

Volume is the amount of space inside a three-dimensional object: how much water a tank holds, how much concrete a slab needs, or how many cubic feet a shipping crate contains. On WorkKeys Applied Math you will use two volume formulas most often, the rectangular prism (box) and the cylinder. The formula sheet supplies both. The harder part of these items is the second step, converting cubic feet into gallons, cubic yards, or a material quantity you can order.

The core formulas and conversions

The single most common error is mixing units. If a tank depth is given in inches while length and width are in feet, convert everything to feet first. A depth of 6 inches is 6 / 12 = 0.5 ft. Do that conversion before you multiply, not after.

Rectangular prisms: tanks and crates

Water tank example. A rectangular tank is 4 ft long, 3 ft wide, and 5 ft deep. What is its volume in cubic feet, and how many gallons of water will it hold when full?

Volume = L x W x H = 4 x 3 x 5 = 60 cubic feet. Gallons = 60 x 7.48 = 448.8 gallons. Round to about 449 gallons if the answer choices are whole numbers.

Shipping crate example. A crate measures 2 ft by 2 ft by 3 ft. How many cubic feet of packing space does it provide?

Volume = 2 x 2 x 3 = 12 cubic feet. If each product box occupies 0.5 cubic feet, the crate holds 12 / 0.5 = 24 product boxes, assuming they pack together without wasted space. Notice that the order in which you multiply the three edges does not matter: 2 x 2 x 3, 2 x 3 x 2, and 3 x 2 x 2 all equal 12. That freedom lets you multiply the two easiest numbers first and then the third.

Cylinders: drums and pipes

For a cylinder you need the radius of the circular end. Watch for problems that hand you a diameter instead. Halve it first, then square the radius. A cylinder is really a stack of identical circles, so its volume is the area of one circular end (pi x r x r) multiplied by the height h, which is why the radius is squared but the height is not.

Cylinder tank example. A cylindrical tank has a radius of 2 ft and a height of 6 ft. Using pi = 3.14, find the volume and the gallons it holds.

Volume = pi x r x r x h = 3.14 x 2 x 2 x 6 = 3.14 x 24 = 75.36 cubic feet. Gallons = 75.36 x 7.48 = 563.69 gallons, about 564 gallons.

Oil drum example. A drum has a diameter of 2 ft and a height of 3 ft. Find its volume.

First convert diameter to radius: r = 2 / 2 = 1 ft. Volume = pi x r x r x h = 3.14 x 1 x 1 x 3 = 9.42 cubic feet. In gallons that is 9.42 x 7.48 = 70.46 gallons. Trust the arithmetic here rather than what you expect a drum to hold, because the numbers are simplified for practice.

Converting volume to a material order

Concrete slab example. A parking pad is 30 ft long, 20 ft wide, and 6 inches thick. Concrete is ordered in cubic yards. How many cubic yards are needed?

Convert thickness: 6 inches = 6 / 12 = 0.5 ft. Volume = L x W x H = 30 x 20 x 0.5 = 300 cubic feet. Convert to cubic yards: 300 / 27 = 11.11 cubic yards. Because concrete is delivered in whole yards and you never want to run short, round up to 12 cubic yards.

Gravel example. A trench is 40 ft long, 2 ft wide, and 1.5 ft deep, to be filled with gravel. Gravel is sold by the cubic yard. How much is needed?

Volume = 40 x 2 x 1.5 = 120 cubic feet. Cubic yards = 120 / 27 = 4.44, so order 5 cubic yards. Rounding up ensures the trench is filled even after the gravel settles.

Test tips

• Keep every dimension in the same unit before you multiply. Convert inches to feet by dividing by 12.
• For cylinders, confirm whether the problem gives a radius or a diameter, and halve a diameter first.
• Use 7.48 gallons per cubic foot to convert volume to liquid capacity, and 27 cubic feet per cubic yard for bulk materials.
• For any orderable material such as concrete or gravel, round up to the next whole unit so you do not run short.
• Estimate before finalizing: a box a few feet on each side holds tens of gallons, so an answer in the thousands signals an error.