Practice Maths

Volume of Prisms

Key Ideas

Key Terms

prism
A 3D solid with two identical parallel bases connected by rectangular lateral faces; its cross-section is constant along its length.
cross-section
The shape formed by cutting through a prism parallel to its bases; the same all the way through the prism.
length
The dimension of a prism measuring the distance between its two bases.
rectangular prism (cuboid)
A prism with a rectangular cross-section; V = l × w × h.
triangular prism
A prism with a triangular cross-section; V = ½bh × l.
trapezoidal prism
A prism with a trapezoidal cross-section; V = ½(a + b)h × l.
composite prism
A solid made up of two or more prisms joined together; find the volume of each part and add them.
Hot Tip The cross-section (base) must be the same shape all the way through the prism. Identify the cross-section first, calculate its area, then multiply by the length (the dimension going through the prism).

Worked Example

Rectangular prism: length = 4 cm, width = 3 cm, height = 5 cm
V = l × w × h = 4 × 3 × 5 = 60 cm³

Triangular prism: triangle base = 3 cm, triangle height = 4 cm, prism length = 8 cm
Atriangle = ½ × 3 × 4 = 6 cm²
V = 6 × 8 = 48 cm³

What Is a Prism?

A prism is a 3D solid with two identical, parallel faces called bases, connected by rectangular (or parallelogram) lateral faces. The key feature of a prism is that its cross-section — the shape you get when you cut through it parallel to the bases — is the same all the way through.

Examples of prisms: a rectangular prism (box shape — cross-section is a rectangle), a triangular prism (like a Toblerone box — cross-section is a triangle), a hexagonal prism (cross-section is a hexagon). A cylinder is sometimes treated like a prism with a circular cross-section.

To identify the cross-section of a prism, picture slicing through it parallel to the ends (the bases). The shape of that slice is the cross-section.

The Volume Formula for a Prism

The volume of any prism is given by:

V = A × h

where A is the area of the cross-section (the base) and h is the height (or length) of the prism — the distance between the two bases.

Think of it this way: you are stacking identical cross-sectional layers on top of each other. Each layer has area A. If you stack them to a height of h, the total volume is A × h. This formula works for any prism, regardless of the shape of the cross-section.

Rectangular Prisms

For a rectangular prism (also called a cuboid), the cross-section is a rectangle with area = length × width. So:

V = l × w × h

Example: A box is 8 cm long, 5 cm wide, and 3 cm tall. V = 8 × 5 × 3 = 120 cm3.

Note: it doesn't matter which face you call the "base" — multiplying all three dimensions gives the same answer.

Triangular Prisms

For a triangular prism, the cross-section is a triangle. The area of the triangle is A = 12 × base × height (of the triangle, not the prism!).

Example: A triangular prism has a triangular cross-section with base 6 cm and height 4 cm. The prism is 10 cm long. Area of cross-section = 12 × 6 × 4 = 12 cm2. Volume = 12 × 10 = 120 cm3.

Be careful to use the height of the triangle (the perpendicular height) and the length of the prism separately — do not confuse them.

Using Correct Units

Volume is always expressed in cubic units. If all measurements are in cm, the answer is in cm3. If all measurements are in m, the answer is in m3. If measurements are given in different units, convert all to the same unit first before calculating.

Example: A prism has cross-sectional area 2.5 m2 and height 40 cm. Convert 40 cm to 0.4 m. Volume = 2.5 × 0.4 = 1 m3.

Never mix units within a calculation — this is one of the most common errors in volume problems.

Key tip: The hardest part of prism volume questions is correctly identifying which face is the cross-section and calculating its area. Always look for the shape that stays constant when you "slide" along the length of the prism. Once you have the cross-sectional area, everything else is straightforward: just multiply by the length. Draw the cross-section separately and label it if you are unsure.

Mastery Practice

  1. Find the volume of each rectangular prism (cuboid). Fluency

    1. l = 5 cm, w = 4 cm, h = 3 cm
    2. l = 8 m, w = 2 m, h = 3 m
    3. l = 6 cm, w = 6 cm, h = 6 cm
    4. l = 10 m, w = 4 m, h = 2.5 m
    5. l = 7 cm, w = 3 cm, h = 4 cm
    6. l = 12 mm, w = 5 mm, h = 8 mm
    7. l = 9 m, w = 3 m, h = 2 m
    8. l = 4.5 cm, w = 4 cm, h = 2 cm

  2. Find the volume of each triangular prism. The cross-section is a right-angled triangle unless stated otherwise. Fluency

    1. Triangle: base = 4 cm, height = 3 cm; prism length = 10 cm
    2. Triangle: base = 6 m, height = 5 m; prism length = 8 m
    3. Triangle: base = 8 cm, height = 6 cm; prism length = 12 cm
    4. Triangle: base = 5 mm, height = 4 mm; prism length = 9 mm
    5. Triangle: base = 10 m, height = 7 m; prism length = 4 m
    6. Triangle: base = 3 cm, height = 3 cm; prism length = 20 cm
    7. Triangle: base = 12 cm, height = 5 cm; prism length = 6 cm
    8. Triangle: base = 9 m, height = 4 m; prism length = 5 m

  3. Find the volume of each prism. Fluency

    1. Trapezoidal prism: parallel sides of trapezium = 6 cm and 10 cm, height of trapezium = 4 cm, prism length = 8 cm
    2. Trapezoidal prism: parallel sides = 3 m and 7 m, height = 5 m, prism length = 12 m
    3. L-shaped cross-section: the cross-section can be split into two rectangles, 4 cm × 2 cm and 2 cm × 3 cm; prism length = 10 cm
    4. Trapezoidal prism: parallel sides = 5 mm and 9 mm, height = 6 mm, prism length = 15 mm
    5. L-shaped cross-section: two rectangles, 6 m × 2 m and 3 m × 4 m; prism length = 5 m
    6. Trapezoidal prism: parallel sides = 8 cm and 14 cm, height = 5 cm, prism length = 9 cm
    7. L-shaped cross-section: two rectangles, 5 cm × 3 cm and 4 cm × 2 cm; prism length = 7 cm
    8. Trapezoidal prism: parallel sides = 4 m and 10 m, height = 3 m, prism length = 6 m

  4. Find the missing dimension in each prism. Understanding

    1. Rectangular prism: V = 120 cm³, l = 6 cm, w = 4 cm. Find h.
    2. Rectangular prism: V = 360 m³, l = 9 m, h = 5 m. Find w.
    3. Triangular prism: V = 84 cm³, triangle base = 6 cm, triangle height = 4 cm. Find the prism length.
    4. Rectangular prism: V = 210 mm³, w = 7 mm, h = 5 mm. Find l.
    5. Triangular prism: V = 180 m³, prism length = 10 m, triangle height = 6 m. Find the triangle base.
    6. Rectangular prism: V = 432 cm³, l = 12 cm, w = 6 cm. Find h.
    7. Triangular prism: V = 90 cm³, prism length = 9 cm, triangle base = 5 cm. Find the triangle height.
    8. Rectangular prism: V = 96 m³, l = w and h = 4 m. Find l (and w).

  5. Find the volume of each prism, then convert to litres. Understanding

    1. Rectangular container: 20 cm × 15 cm × 10 cm
    2. Rectangular fish tank: 50 cm × 30 cm × 40 cm
    3. Triangular prism trough: triangle base = 40 cm, triangle height = 30 cm, trough length = 80 cm
    4. Rectangular box: 25 cm × 20 cm × 8 cm
    5. Rectangular paddling pool: 1.2 m × 0.8 m × 0.3 m (convert m to cm first)
    6. Rectangular planter box: 60 cm × 25 cm × 20 cm
    7. Triangular prism vessel: triangle base = 24 cm, triangle height = 20 cm, length = 50 cm
    8. Rectangular storage bin: 45 cm × 35 cm × 28 cm

  6. Solve each problem, showing all working and units at each step. Problem Solving

    1. A packaging company needs to design a rectangular box to hold exactly 1 litre of juice.
      1. What volume in cm³ does the box need to hold?
      2. The box must be 10 cm long and 5 cm wide. What height is required?
      3. If the height is rounded up to the nearest whole cm, what is the actual capacity of the box in litres?
    2. A garden water feature is a triangular prism trough. The triangular cross-section has a base of 60 cm and a height of 45 cm. The trough is 1.5 m long.
      1. Find the area of the triangular cross-section in cm².
      2. Convert the trough length to cm.
      3. Calculate the volume in cm³ and the capacity in litres.
    3. A concrete slab for a garden path is rectangular: 3 m long, 1.2 m wide, and 0.1 m thick.
      1. Calculate the volume of concrete needed in m³.
      2. Convert this to cm³.
      3. Concrete costs $180 per m³. What is the cost of the concrete for this slab?

  7. Find the missing dimension using the volume formula. Problem Solving

    Reverse problems. Rearrange V = Ah to find A or h: h = V ÷ A. Then rearrange the area formula to find a missing dimension of the cross-section.
    1. A trapezoidal prism has volume 840 cm³, prism length 12 cm, and a trapezium with parallel sides 8 cm and 12 cm. Find the height of the trapezium.
    2. A triangular prism has volume 270 m³. The prism length is 15 m and the triangular cross-section has a base of 9 m. Find the height of the triangle.
    3. A rectangular prism has volume 1.5 m³. Its height is 0.5 m and its width is 1.5 m. Find its length.
    4. A triangular prism-shaped ramp has a volume of 3.6 m³. The ramp is 4 m long and the triangular cross-section is a right-angled triangle with a base of 1.8 m. Find the height of the triangle.

  8. Find the volume of each composite prism (two prisms joined). Problem Solving

    Composite volumes. Split the solid into recognisable prisms, find each volume separately, then add them.
    1. A solid is made of a rectangular prism (10 cm × 4 cm × 3 cm) sitting on top of another rectangular prism (10 cm × 8 cm × 5 cm). Find the total volume.
    2. A cross-shaped prism has a cross-section that can be split into three rectangles: a horizontal bar (8 cm × 2 cm) and a vertical bar (2 cm × 6 cm, excluding overlap). The prism is 5 cm long. Find the total volume.
    3. A solid is a triangular prism (triangle base = 6 m, triangle height = 4 m, prism length = 10 m) sitting on a rectangular prism (6 m × 4 m × 10 m). Find the total volume.
    4. An L-shaped prism has a cross-section consisting of two rectangles: 7 cm × 3 cm and 4 cm × 2 cm. The prism is 8 cm long. Calculate the total volume.

  9. Volume problems involving design and cost. Problem Solving

    Real-world context. Volume problems often involve finding cost, comparing options, or checking whether a container meets requirements.
    1. A company manufactures rectangular boxes with dimensions 25 cm × 18 cm × 12 cm.
      1. Find the volume of one box in cm³.
      2. Convert the volume to litres.
      3. How many boxes are needed to store 200 litres of goods (assuming boxes can be fully packed)?
    2. A triangular prism tent has a triangular cross-section with base 2.4 m and height 1.8 m. The tent is 3.5 m long.
      1. Find the volume of air inside the tent in m³.
      2. A person needs at least 3 m³ of air space. Can two people comfortably sleep in this tent?
    3. Sand is sold in bags. Each bag has dimensions 60 cm × 30 cm × 10 cm (a rectangular prism).
      1. Find the volume of each bag in cm³ and in litres.
      2. A garden bed requires 300 L of sand. How many bags are needed?

  10. Multi-step volume problems. Problem Solving

    Multi-step. Break the problem into steps: identify the shape, write the formula, substitute, calculate, and convert units as needed.
    1. A swimming lane is a rectangular prism 50 m long, 2.5 m wide, and 2 m deep. Water is pumped in at 5000 L per minute. How long does it take to fill one lane? Give your answer in hours and minutes.
    2. A chocolate bar is a triangular prism: the triangular cross-section has base 3 cm and height 2 cm, and the bar is 15 cm long. A box holds 24 such bars in a single layer. What is the minimum volume of the box needed to hold all 24 bars?
    3. A warehouse has floor dimensions 20 m × 15 m and a ceiling height of 6 m.
      1. Find the volume of the warehouse in m³.
      2. Rectangular storage pallets each measure 1.2 m × 1 m × 2 m. How many pallets can fit in the warehouse, assuming no stacking?
    4. A trapezoidal prism-shaped water channel has parallel sides of 0.8 m and 1.2 m, a height of 0.5 m, and a length of 50 m. Water flows through at 0.2 m³ per second. How many minutes does it take to fill the channel completely?
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