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\[ V_{cube} = x^3 \qquad V_{cuboid} = l \times b \times h \qquad V_{cylinder} = \pi r^2 h \]
Welcome to Grade 8 Volume. Volume is the amount of space a 3D shape takes up, measured in cubic units (mm³, cm³, m³). In this lesson you will learn to calculate the volume and surface area of cubes, cuboids and cylinders, and the volume of prisms.
\[ 1\text{ cm}^3 = 1\text{ cm} \times 1\text{ cm} \times 1\text{ cm} \]
Volume measures how much space a solid (3D) shape fills. Just as area is measured in square units, volume is measured in cubic units: mm³, cm³ or m³. One cm³ is the space taken up by a tiny cube 1 cm on every side.
A cube has all sides equal in length. If a side is \(x\), its volume is \(x\) multiplied by itself three times:
\( V = x \times x \times x = x^3 \)
Its surface area is the total area of all 6 identical square faces: \( SA = 6x^2 \).
\[ V = 5^3 = 125 \text{ cm}^3 \qquad SA = 6(5^2) = 150 \text{ cm}^2 \]
A cube has sides of 5 cm. \(V = x^3 = 5^3 = 125\) cm³. Its surface area is \(SA = 6x^2 = 6 \times 25 = 150\) cm² — six identical square faces, each 5 cm × 5 cm.
A cuboid (rectangular box) has three different edge lengths: length, breadth and height. Its volume is all three multiplied together:
\( V = l \times b \times h \)
Its surface area adds up all 3 pairs of identical rectangular faces: \( SA = 2(lb + lh + bh) \).
Find the volume and surface area of this cuboid (4 m × 18 m × 5 m).
\( V = 4 \times 18 \times 5 = 360 \text{ m}^3 \)
\( SA = 2(4{\times}18) + 2(4{\times}5) + 2(5{\times}18) = 144 + 40 + 180 = 364 \text{ m}^2 \)
\[ V = 2 \times 1{,}5 \times 1 = 3 \text{ m}^3 = 3\,000 \text{ litres} \]
A rectangular water tank is 2 m long, 1,5 m wide and 1 m high. \(V = 2 \times 1{,}5 \times 1 = 3\) m³. Since 1 m³ = 1 000 litres, the tank holds \(3 \times 1\,000 = 3\,000\) litres of water.
\[ V_{cube} = x^3 \qquad\qquad V_{cuboid} = l \times b \times h \]
A cube's volume is its side cubed. A cuboid's volume is length × breadth × height. Surface area adds up the area of every face.
A cylinder is like a circular prism — a circle stretched into a tube. Its volume is the area of the circular base (\(\pi r^2\)) multiplied by its height:
\( V = \pi r^2 h \)
Its total surface area adds the curved side (\(2\pi rh\)) plus the two circular ends (\(2\pi r^2\)): \( SA = 2\pi rh + 2\pi r^2 \).
Find the volume and total surface area of this cylinder (radius 4 cm, height 6 cm). Use \(\pi \approx 3{,}14\).
\( V = \pi r^2 h = 3{,}14 \times 4^2 \times 6 = 3{,}14 \times 96 = 301{,}44 \text{ cm}^3 \)
Curved surface: \( 2\pi rh = 2 \times 3{,}14 \times 4 \times 6 = 150{,}72 \text{ cm}^2 \)
Two ends: \( 2\pi r^2 = 2 \times 3{,}14 \times 16 = 100{,}48 \text{ cm}^2 \)
Total: \( SA = 150{,}72 + 100{,}48 = 251{,}2 \text{ cm}^2 \)
\[ V = \pi r^2 h \qquad\qquad SA = 2\pi rh + 2\pi r^2 \]
A cylinder's volume is its circular base area times its height. If you're only given the diameter, halve it to get the radius first — the same rule as with circles.
A prism is any 3D shape with a uniform cross-section — the same flat shape all the way through, like a loaf of bread. Its volume is simply the area of that cross-section (\(A\)) multiplied by the prism's length:
\( V = A \times l \)
This works no matter what the cross-section shape is — triangle, pentagon, even a circle (that's a cylinder!).
Find the volume of this triangular prism (triangular cross-section: base 8 cm, height 6 cm; prism length 10 cm).
Area of triangular end: \( A = \tfrac{1}{2}(8 \times 6) = 24 \text{ cm}^2 \)
Volume: \( V = A \times l = 24 \times 10 = 240 \text{ cm}^3 \)
\[ V_{prism} = A_{cross\text{-}section} \times \text{length} \]
Whatever the cross-section shape, find its area first, then multiply by the prism's length. A cuboid is just a prism with a rectangular cross-section — a cylinder is a prism with a circular one.
\[ \begin{aligned} V_{cube}&=x^3 & V_{cuboid}&=l\times b\times h \\ V_{cylinder}&=\pi r^2 h & V_{prism}&=A\times l \end{aligned} \]
You now know how to:
• Calculate the volume of a cube: \(V = x^3\)
• Calculate the volume of a cuboid: \(V = l \times b \times h\)
• Calculate the volume of a cylinder: \(V = \pi r^2 h\)
• Calculate the volume of any prism: \(V = A \times l\)
• Calculate the surface area of cubes, cuboids and cylinders
Ready to test yourself?