Surface Area
Section: Geometrical Reasoning, Shapes and Measurements | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
What is Surface Area?
Surface area is the total area of every face of a 3D shape, added together. A net - the shape unfolded flat - is a useful way to see and calculate every face.
Surface Area of a Cube and Cuboid
A cube has 6 identical square faces: SA = 6 × (side length)². A cuboid has 3 pairs of identical rectangular faces: SA = 2(lw + lh + wh).
-
Question: Find the surface area of a cube with side length 5 cm.
- Step 1: SA = 6 × 5² = 6 × 25
- Answer: 150 cm²
-
Question: Find the surface area of a cuboid 8 cm × 4 cm × 3 cm.
- Step 1: Find each pair's area: lw = 32, lh = 24, wh = 12
- Step 2: Add them: 32 + 24 + 12 = 68
- Step 3: Double, since each face has a matching pair: 68 × 2
- Answer: 136 cm²
Surface Area of a Triangular Prism
A triangular prism has 2 identical triangular ends and 3 rectangular side faces - one for each side of the triangle. SA = 2 × (triangle area) + (perimeter of triangle × length of prism).
-
Question: A triangular prism has a right-angled triangular cross-section with legs 3 cm and 4 cm (hypotenuse 5 cm), and is 10 cm long. Find its surface area.
- Step 1: Triangle area = ½ × 3 × 4 = 6 cm². Two triangles = 12 cm²
- Step 2: Triangle perimeter = 3 + 4 + 5 = 12 cm
- Step 3: Rectangular faces = 12 × 10 = 120 cm²
- Step 4: Total = 12 + 120
- Answer: 132 cm²
Surface Area of a Cylinder
A cylinder has 2 circular ends and 1 curved surface, which unrolls into a rectangle with width equal to the circumference. SA = 2πr² + 2πrh.
A cylinder's net: two circles plus a rectangle whose width is the circumference
-
Question: Find the surface area of a cylinder with radius 4 cm and height 10 cm (use π ≈ 3.14).
- Step 1: Two circles: 2πr² = 2 × 3.14 × 16 = 100.48 cm²
- Step 2: Curved surface: 2πrh = 2 × 3.14 × 4 × 10 = 251.2 cm²
- Step 3: Add: 100.48 + 251.2
- Answer: 351.68 cm²
Surface Area of a Pyramid
A pyramid's surface area is its base area plus the areas of its triangular side faces - using each face's slant height, not the pyramid's vertical height.
-
Question: A square-based pyramid has a base of side 6 cm, and each triangular face has a slant height of 5 cm. Find its surface area.
- Step 1: Base area = 6 × 6 = 36 cm²
- Step 2: Each triangular face = ½ × 6 × 5 = 15 cm². Four faces = 60 cm²
- Step 3: Total = 36 + 60
- Answer: 96 cm²
Real-World Applications
Surface area calculations matter in many practical situations:
- Gift wrapping: the paper needed to cover a box.
- Painting: the amount of paint needed for a cylindrical tank.
- Tents: the fabric needed for a triangular-prism or pyramid-shaped tent.
- Packaging design: minimising material for a given volume.
- Roofing: the materials needed for a pyramid-shaped roof.
Exam Tips
Common Mistakes
MistakeForgetting to double a face's area when the shape has a matching pair, e.g. only counting one triangular end of a prism
Fixidentify ALL faces first - most prisms have pairs of matching faces that both need counting
MistakeUsing a pyramid's vertical height instead of its slant height for the triangular faces
Fixthe triangular side faces of a pyramid always use the slant height
MistakeConfusing surface area with volume
Fixsurface area is a 2D measure (units²); volume is a 3D measure (units³)
MistakeForgetting the curved surface of a cylinder and only calculating the two circular ends
Fixa cylinder's surface area needs the two circles AND the curved surface (2πrh)
For Exams
- Sketch or imagine the net of a 3D shape to make sure you've counted every face.
- Learn each formula: cube/cuboid, triangular prism, cylinder, pyramid.
- For pyramids: use slant height for the triangular faces, not vertical height.
- Check the question: surface area (units²) or volume (units³)?
Interactive revision notes, videos and practice questions load below.