Volume of a rectangular prism: V = l × w × h

What volume means

Volume is the amount of space inside a solid shape, and it is measured in cubes. A cube 1 cm along each edge has a volume of 1 cubic centimeter, written 1 cm³. In the US, cubic inches (in³) and cubic feet (ft³) are common too.

A rectangular prism is a box shape. It has six rectangular faces, and every corner is a right angle. A cube is a rectangular prism with all its edges the same length. In England, the same shape is called a cuboid.

Why volume is length × width × height

Build a prism 4 cubes long, 3 cubes wide and 2 cubes high. The bottom layer is a 4 by 3 rectangle of cubes, so it holds 4 × 3 = 12 cubes. There are 2 layers, so the prism holds 12 × 2 = 24 cubes. Its volume is 24 cubic units.

That is all the formula says. Length × width gives the cubes in one layer, which is the area of the base, B. Multiplying by the height counts the layers. So V = l × w × h, or V = B × h. You can multiply the three edges in any order and get the same answer.

Volume examples

These examples get a little harder each time.

• A cube with edges of 5 in: V = 5 × 5 × 5 = 125 in³.
• A cereal box 8 in long, 3 in wide and 12 in high: V = 8 × 3 × 12 = 288 in³.
• A missing edge: a prism has a volume of 168 cm³, a length of 7 cm and a width of 4 cm. The base area is 7 × 4 = 28 cm², so the height is 168 ÷ 28 = 6 cm.
• From the base area: a tank with a base area of 30 ft² holds 120 ft³ of water when full. The water is 120 ÷ 30 = 4 ft deep.
• Fraction edges: a box 2½ m by 2 m by 1½ m has a volume of 5/2 × 2 × 3/2 = 15/2 = 7½ m³.

Shapes made from two prisms

Volume adds up. If a solid is made of two rectangular prisms that do not overlap, find the volume of each and add them. A step made from a 6 ft by 4 ft by 2 ft block with a 3 ft by 4 ft by 3 ft block beside it has a volume of 48 + 36 = 84 ft³.

When a piece is cut out, subtract instead. A 10 cm by 4 cm by 5 cm block with a 3 cm by 4 cm by 2 cm notch cut from one end has 200 − 24 = 176 cm³ left.

Packing with fraction cubes

In grade 6, edges can be fractions. A box 2 cm by 1½ cm by 1 cm can be packed with cubes ½ cm on each edge: 4 fit along the length, 3 along the width and 2 up the height, so it holds 4 × 3 × 2 = 24 small cubes. Eight small cubes make one 1 cm cube, so the volume is 24 ÷ 8 = 3 cm³. The formula agrees: 2 × 1½ × 1 = 3 cm³.

How it is taught in school

In the US, Common Core introduces volume in grade 5. Students learn that volume is measured in unit cubes (5.MD.C.3), measure it by counting cubes (5.MD.C.4), and connect that to the formulas V = l × w × h and V = B × h, including the volume of shapes made from two rectangular prisms (5.MD.C.5). Grade 6 extends this to edges with fraction lengths (6.G.A.2), and grade 7 to other prisms (7.G.B.6).

In England, Year 5 pupils estimate volume, for example by building cuboids from 1 cm³ blocks. Year 6 pupils calculate, estimate and compare the volume of cubes and cuboids in cm³ and m³, and learn when a formula for volume can be used.

How parents can help

• Build prisms from sugar cubes, dice or building blocks. Count one layer, then the layers, before showing the formula.
• Measure real boxes at home, such as a cereal box or a shoe box, and work out their volumes.
• Ask which unit the answer needs. Volume is always in cubic units: cm³, in³ or ft³.
• Link volume to capacity: 1,000 cm³ holds 1 liter, so a 30,000 cm³ fish tank holds 30 liters.
• Keep sessions short. Print a fresh worksheet, let your child mark it with the answer key, and go through the worked steps for any they missed.