How to find volume: rectangular prisms, cubes and cylinders

Your fifth grader is packing a moving box with toy blocks and asks how many will fit. That question is volume, and the answer comes from three measurements and two multiplications. Learning how to find volume starts with a box and a pile of small cubes, and by the end of grade 5 most children can find the volume of a rectangular prism from a drawing alone.

What is volume?

Volume is the amount of space inside a solid shape. OpenStax Prealgebra describes the space inside a crate as "the volume, a cubic measure", and that crate is a good model for every box in your kitchen.

A cereal box, a fish tank and a shoebox all have volume, because each one has length, width and height. A sheet of paper has area but almost no volume, because it has almost no height. The guide to area and perimeter covers the flat measurements; volume adds the third direction.

Why volume uses cubic units

Volume uses cubic units because it counts how many small cubes fill a shape. Common Core standard 5.MD.C.3 says that "a cube with side length 1 unit, called a 'unit cube,' is said to have 'one cubic unit' of volume", and that a solid "packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units".

A cube 1 cm on each side is one cubic centimeter, written 1 cm³. Area counts squares (cm²), and volume counts cubes (cm³). The small raised 3 means three lengths were multiplied, and the guide to exponents explains why "cubed" and the raised 3 mean the same thing.

Standard 5.MD.C.4 asks children to "measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units", so the first lessons are hands-on counting.

How to find the volume of a rectangular prism, step by step

The volume of a rectangular prism is length × width × height, written V = l × w × h. Standard 5.MD.C.5b asks children to "apply the formulas V = l × w × h and V = b × h for rectangular prisms", where b is the area of the base. Take a box 6 cm long, 4 cm wide and 3 cm tall.

1. Measure the three edges. Write down the length (6 cm), the width (4 cm) and the height (3 cm). Any edge can be the length; the answer stays the same.
2. Put every edge in the same unit. A box 2 m long and 50 cm wide needs 200 cm and 50 cm before you multiply.
3. Find the area of the base. Multiply length × width: 6 × 4 = 24 cm². This is how many cubes cover the bottom of the box in one layer.
4. Multiply by the height. Three layers of 24 cubes make 24 × 3 = 72. This is the formula V = B × h, base area × height.
5. Check with the other order. Multiply 4 × 3 = 12 first, then 12 × 6 = 72. The product does not change, which is the associative property standard 5.MD.C.5a mentions.
6. Write the answer in cubic units. The volume is 72 cm³, or 72 cubic centimeters.

Larger boxes use the same steps, with bigger multiplications. A moving box 18 inches long, 18 inches wide and 16 inches tall holds 18 × 18 × 16 = 5,184 cubic inches, and the guide on how to teach long multiplication helps when the edges reach two digits.

Free printable worksheetMultiplication — 5th GradersWorksheet · PDF

The part people get wrong

The most common wrong answer adds the edges instead of multiplying them. A child who writes 6 + 4 + 3 = 13 has found a length, and a box that holds 13 cubes would be far smaller than the drawing. Ask "how many cubes in one layer, and how many layers?" and the multiplication follows.

The second mistake is mixing units. A tank 2 m long, 50 cm wide and 40 cm tall is 200 × 50 × 40 = 400,000 cm³, which is 400 liters. Multiplying 2 × 50 × 40 = 4,000 gives a number with no unit at all.

How to find volume for other shapes: formulas side by side

How to find volume for other solids follows the same idea as the box: find the area of the base, then multiply by the height. OpenStax gives the cube as V = s³, the cylinder as V = π r² h or V = B h, the cone as one third of a cylinder, and the sphere as V = 4/3 π r³.

The cylinder and the cone in the table share a radius of 3 cm and a height of 10 cm, so the cone holds one third of what the cylinder holds. The values use π ≈ 3.14, so each answer is close to the exact value. The guide to multiplying and dividing decimals covers the 3.14 × 90 step.

Which volume formula your child needs at which grade

The volume formula a child needs depends on the grade, and US schools introduce them in a clear order. Grade 5 covers rectangular prisms with whole-number edges (5.MD.C.5). Grade 6 moves to fractional edges: standard 6.G.A.2 asks children to "find the volume of a right rectangular prism with fractional edge lengths".

Grade 7 adds other right prisms (7.G.B.6), and grade 8 asks students to "know the formulas for the volumes of cones, cylinders, and spheres" (8.G.C.9).

A grade 5 child does not need π yet. The overview of what math each grade should know shows where volume sits next to fractions and decimals.

Kitchen-table volume activities

Kitchen-table volume activities turn the formula into something a child can hold. Each one takes ten minutes and uses things you already own.

• Build the box from cubes: Use base-ten unit cubes, snap cubes or sugar cubes. Build one layer 4 cubes by 3 cubes, count 12, then stack layers and count by twelves. Sugar cubes are an "improvised unit" in the words of 5.MD.C.4, so the answer is in sugar cubes, not cm³.
• Measure a cereal box: A box 30 cm tall, 20 cm wide and 7 cm deep holds 30 × 20 × 7 = 4,200 cm³. Measure yours and compare it with a smaller box from the pantry.
• Fill a container with water: Measure a plastic food container, predict its volume, then pour water in from a measuring jug and read how many milliliters it took.
• Compare two boxes: Ask which of two boxes holds more before measuring. A tall, narrow box often fools the eye.

Liters, milliliters and cubic centimeters

Liters and cubic centimeters measure the same thing, so a measuring jug checks a calculation. The National Institute of Standards and Technology states that "1 cm3 = 1 milliliter (mL)" and "1000 cm3 = 1 cubic decimeter (dm3)", which is 1 liter.

A container 10 cm × 10 cm × 10 cm holds 1,000 cm³, so it holds exactly 1 liter of water. A fish tank 60 cm long, 30 cm wide and 40 cm tall holds 60 × 30 × 40 = 72,000 cm³, which is 72 liters. Children who see the jug empty into the box remember the conversion far longer than a rule on a worksheet.

How to check progress with volume

Five short questions show whether a child understands volume or is repeating a formula.

• The sorting question: Ask whether filling a pool, painting a wall and wrapping a gift are area or volume jobs. Filling the pool is volume; the wall and the wrapping paper are area.
• The basic box: Give a box 5 by 3 by 2. The volume is 30 cubic units, and the child should name the unit.
• The layer question: Show a box 4 cubes by 5 cubes with 3 layers. A child who says "20 in a layer, so 60" understands V = B × h.
• The missing edge: Give a volume of 60 cm³, a length of 5 cm and a width of 4 cm. The height is 60 ÷ 20 = 3 cm.
• The conversion: Ask how many liters a 20 cm × 10 cm × 10 cm box holds. The volume is 2,000 cm³, which is 2 liters.

A five-minute kitchen-table test

A five-minute volume test mixes three question types: one box to calculate, one missing edge and one liter conversion. Ask the child to say "layer, then height" out loud before multiplying.

A child who knows the method but stalls on 6 × 4 or 24 × 3 needs the multiplication facts more than the geometry. The times tables game drills those facts in the browser, and the math game for 5th graders covers the grade 5 number work every volume problem leans on.

Play free in your browserTimes TablesGrade 3+

Common volume mistakes and edge cases

Four mistakes cause most wrong volume answers, and each has a short fix.

• Adding the edges: A 6 × 4 × 3 box gets 13 instead of 72. Ask "how many in one layer?" before any calculation.
• Using area units: An answer of 72 cm² describes a flat shape. Volume always uses cubic units: cm³, in³ or ft³.
• Mixing units: A box 1 m by 20 cm by 10 cm is 100 × 20 × 10 = 20,000 cm³, or 20 liters. Convert every edge before you multiply.
• Missing a part of a composite shape: An L-shaped solid is two boxes. Standard 5.MD.C.5c asks children to "find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts". A step 5 × 3 × 2 = 30 cubes next to a block 2 × 3 × 4 = 24 cubes makes 30 + 24 = 54 cubes.

Edge case: doubling every edge multiplies volume by 8

Doubling every edge of a box multiplies its volume by 8, because the 2 appears three times: 2 × 2 × 2 = 8. A box 3 by 1 by 2 holds 6 cubes, and a box 6 by 2 by 4 holds 48 cubes, which is 8 times as many.

Area grows by 4 when the sides double, and volume grows by 8. A box 2 × 3 × 4 = 24 becomes 4 × 6 × 8 = 192, and 192 ÷ 24 = 8. The same rule holds for a cooking pot: twice as wide and twice as tall means 8 times as much soup.

Your volume checklist and next steps

• Three edges: measure the length, the width and the height of every box.
• One unit: convert every edge to the same unit before multiplying.
• Layer, then height: find the base area, then multiply by the number of layers.
• Cubic units: write cm³, cubic inches or cubic feet on every answer.
• Liters: divide cm³ by 1,000 to get liters, and check with a measuring jug.
• Composite shapes: split an L-shaped solid into two boxes and add the volumes.

MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty and a short daily practice that builds the multiplication facts behind every volume problem. Children can also challenge 1 to 4 friends or family members in real-time math battles, in any of 33 languages.