Practice Maths

L23 — Introduction to Prisms & Volume

Key Terms

prism
A 3D solid with two identical, parallel bases (the cross-section) connected by rectangular faces. Named by the shape of its base.
cross-section
The 2D shape you get when a prism is sliced parallel to its base. The cross-section is the same all the way along a prism.
volume
The amount of 3D space inside a solid. Measured in cubic units: mm³, cm³, m³.
cubic units
The units used for volume. A cubic centimetre (cm³) is a 1 cm × 1 cm × 1 cm cube.
capacity
The amount a container can hold. Capacity is linked to volume: 1 cm³ = 1 mL.

Prisms and Their Names

A prism is named by the shape of its base. Each prism has two identical bases and rectangular faces connecting them.

Rectangular prism
Triangular prism
Cube

Volume: Stacking Layers of Unit Cubes

Volume counts how many 1×1×1 cubes fill a solid exactly.

  • Step 1 — Count the cubes in one layer (= length × width).
  • Step 2 — Multiply by the number of layers (the height).
  • Result — Volume = length × width × height

Worked Example — Counting Layers

A solid is 3 cubes long, 2 cubes wide, and 4 cubes high. Find its volume.

Step 1 — One layer: 3 × 2 = 6 cubes.

Step 2 — Four layers: 6 × 4 = 24 cm³.

Cubic vs Square Units: Area is in cm² (flat surface). Volume is in cm³ (3D space). Never write cm² as a volume answer — always cm³, m³, or mm³.
Volume and Capacity: 1 cm³ = 1 mL. So a juice carton with volume 250 cm³ holds exactly 250 mL.

What Is a Prism?

A prism is a 3D solid with a special property: its cross-section is the same all the way along its length. If you sliced through a prism at any point parallel to its base, you would always get the same shape. This uniform cross-section is the defining feature of a prism.

A prism always has:

  • Two identical, parallel bases (the cross-section shape).
  • Rectangular faces connecting the bases.
A cylinder looks like a prism but its base is a circle (a curved shape). Cylinders are not prisms by the strict definition because their connecting “faces” are curved, not flat rectangles. However, the volume method is the same.

Naming Prisms

Every prism is named after the shape of its base:

  • Rectangular prism (cuboid): rectangle base — e.g., a brick, a tissue box, a room.
  • Triangular prism: triangle base — e.g., a Toblerone box, a tent shape.
  • Cube: square base where all three dimensions are equal — a special rectangular prism.
A cube is a special type of rectangular prism where every edge has the same length. Because of this, a cube’s volume formula simplifies to V = s³ (side length cubed).

Volume: Why It Is Measured in Cubic Units

Volume measures 3D space, so its unit must be three-dimensional. A cubic centimetre (cm³) is a tiny cube exactly 1 cm on every edge. If you packed a box with these tiny cubes and needed 120 of them to fill it, the box has a volume of 120 cm³.

The formula comes directly from the layer method:

  • Base area (length × width) gives the number of cubes in one layer.
  • Multiplying by height counts all the layers.
  • Result: V = length × width × height

Choosing the Right Unit

Match the cubic unit to the size of the object:

  • mm³ — very tiny objects (a grain of rice, a raindrop).
  • cm³ — everyday small objects (a dice, a juice carton, a book).
  • m³ — large objects or spaces (a skip bin, a shipping container, a room).

Volume and Capacity

For liquids, we often use millilitres (mL) and litres (L) instead of cubic centimetres. The conversion is exact: 1 cm³ = 1 mL. This means a container with a volume of 500 cm³ holds exactly 500 mL, which is half a litre.

1 L = 1000 mL = 1000 cm³. So to convert cm³ to litres, divide by 1000.

  1. Is It a Prism?

    Each diagram shows a 3D solid. For each one, state whether it is a prism. If it is, name it. If it is not, explain why.

    (a)
    (b)
    (c)
    (d)
    1. Is solid (a) a prism? If yes, name it. How many rectangular faces does it have?
    2. Is solid (b) a prism? The base is a triangle and the solid has 4 faces. Give a reason for your answer.
    3. Is solid (c) a prism? If yes, name it and describe the shape of its cross-section.
    4. Is solid (d) a prism? Explain using the definition of a prism.
    5. A solid has two identical hexagonal bases and six rectangular faces. Name this solid and explain how you know it is a prism.

  2. True or False?

    State whether each statement is true or false. If false, correct it.

    1. A cube is a special type of rectangular prism.
    2. The cross-section of a triangular prism is a rectangle.
    3. Volume is measured in cm².
    4. A prism with a square base and four equal rectangular faces is called a square prism.
    5. A cylinder is a prism because it has two identical circular bases.

  3. Counting Unit Cubes

    Each diagram shows a solid built from 1 cm³ unit cubes. Find the volume of each solid.

    (a) 2×2×2
    (b) 4×3×2
    (c) 2×2×3
    1. Count the unit cubes in solid (a). What is its volume?
    2. Count the unit cubes in solid (b). What is its volume?
    3. Count the unit cubes in solid (c). What is its volume?
    4. Solid (b) has some cubes that cannot be seen from the front. How many hidden cubes are there?
    5. A rectangular solid has 3 cubes in the bottom layer and 4 layers. What is its volume?

  4. Volume Formula Practice

    Use V = l × w × h to find the volume of each rectangular prism.

    1. l = 5 cm, w = 3 cm, h = 2 cm
    2. l = 8 m, w = 4 m, h = 3 m
    3. l = 10 cm, w = 6 cm, h = 5 cm
    4. A cube with side length 4 cm.
    5. l = 12 mm, w = 5 mm, h = 2 mm

  5. The Layer Method

    Answer each part about volume using the layer method.

    1. A layer of unit cubes is 6 cubes long and 4 cubes wide. How many cubes are in one layer?
    2. If the solid from part (a) is 5 layers high, what is the total volume?
    3. A different solid has layers of 20 cubes each. The total volume is 100 cm³. How many layers does it have?
    4. A rectangular prism has a base area of 18 cm² and a height of 7 cm. Find its volume using V = base area × height.

  6. Choosing the Right Unit

    Choose the most appropriate unit (mm³, cm³, or m³) for each volume.

    1. The volume of a juice box.
    2. The volume of a swimming pool.
    3. The volume of a grain of sand.
    4. The volume of a school classroom.

  7. Volume and Capacity

    Use the conversion 1 cm³ = 1 mL and 1 L = 1000 mL.

    1. A water bottle has a volume of 600 cm³. How many mL does it hold?
    2. A fish tank has a volume of 30,000 cm³. How many litres of water does it hold?
    3. A pot holds 3.5 L of soup. What is this volume in cm³?
    4. A container has a volume of 2500 cm³. Is this more or less than 2 litres? Show your reasoning.

  8. Finding a Missing Dimension

    Each rectangular prism has a known volume and two known dimensions. Find the missing dimension.

    1. V = 60 cm³, l = 10 cm, w = 3 cm. Find h.
    2. V = 120 m³, l = 6 m, h = 4 m. Find w.
    3. V = 216 cm³. The solid is a cube. Find the side length.
    4. V = 48 cm³, base area = 16 cm². Find h.

  9. Filling the Pool

    A rectangular swimming pool is 12 m long, 6 m wide, and 2 m deep.

    12 m 2 m 6 m
    1. Calculate the total volume of the pool in m³.
    2. The pool is only filled to a depth of 1.5 m. What volume of water is in the pool?
    3. How many more cubic metres of water are needed to fill the pool completely from a depth of 1.5 m?

  10. Error Analysis and Reasoning

    1. Aisha says a 4 cm × 5 cm × 3 cm box has a volume of 60 cm². Explain her error and write the correct answer. (Hint: think about the unit.)
    2. Two boxes have the same volume of 48 cm³. Box A is 4 cm × 3 cm × 4 cm. Write one set of possible dimensions for Box B (different from Box A). Show your working.
    3. A rectangular prism has length 10 cm, width 2 cm, and height 3 cm. If only the height is doubled, by how much does the volume increase? Explain why this happens.
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