GCSE Maths / Edexcel

Surface area of 3D shapes

Find the total outside area of cuboids, prisms, cylinders and simple compound 3D shapes.

Geometry and MeasuresFoundation and HigherGrades 4 to 7Skill

Curriculum path: GCSE Maths > Edexcel > Geometry and Measures > Surface area

Pearson Edexcel GCSE Maths geometry and measures G16: know and apply formulae for surface area of 3D shapes, including prisms and cylinders.

Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

Surface area means the total area covering the outside of a 3D shape. Imagine wrapping the shape in paper: the amount of paper needed is the surface area.

Surface area is not volume. Volume fills the inside and uses cubic units such as cm³. Surface area covers the outside and uses square units such as cm².

A reliable method is to draw or imagine the net, then find the area of each outside face and add them.

For a cuboid, there are three different face pairs: length x width, length x height, and width x height. Each pair appears twice.

For any prism, find the two matching end faces, then add the rectangles that go around the sides.

For a cylinder, the two circular ends have total area 2πr². The curved face opens into a rectangle with width equal to the circumference 2πr and height h, so its area is 2πrh.

In Edexcel questions, diagrams may hide one face or give a compound solid. Mark the faces carefully so you do not count one twice or miss one.

Key ruleSurface area = add the areas of all outside faces. Cuboid: 2lw + 2lh + 2wh. Cylinder: 2πr² + 2πrh.

Diagram guide

lhlwlhlwwhwhsurface area means all outside facescuboid: 2lw + 2lh + 2wh
Cuboid netA cuboid has six outside faces. Opposite faces are equal, so the three different face areas each appear twice.
trianglesame trianglerectanglessurface area of a prism2 x cross-section area + side rectangles
Prism surface areaA prism has two identical cross-sections plus rectangles around the sides.
rhcurved face2 pi rheight hcylinder: 2 circles + curved rectanglesurface area = 2pi r2 + 2pi rh
Cylinder surface areaA cylinder has two circles and one curved rectangle. The rectangle width is the circumference of the circle.

Worked examples

Cuboid

A cuboid has length 8 cm, width 5 cm and height 3 cm. Find its surface area.

lhlwlhlwwhwhsurface area means all outside facescuboid: 2lw + 2lh + 2wh
Example: cuboid facesFind the three different face areas, then double each one.
  1. length x width = 8 x 5 = 40 cm².
  2. length x height = 8 x 3 = 24 cm².
  3. width x height = 5 x 3 = 15 cm².
  4. Surface area = 2 x 40 + 2 x 24 + 2 x 15 = 158 cm².

Answer: 158 cm²

Triangular prism

A triangular prism has an isosceles triangle cross-section with base 6 cm, equal sides 5 cm and perpendicular height 4 cm. The prism length is 10 cm. Find the surface area.

trianglesame trianglerectanglessurface area of a prism2 x cross-section area + side rectangles
Example: prism facesFind two triangles, then the three side rectangles.
  1. Area of one triangle = 1/2 x 6 x 4 = 12 cm².
  2. Two triangles have area 2 x 12 = 24 cm².
  3. Side rectangles: 6 x 10 = 60, 5 x 10 = 50 and 5 x 10 = 50.
  4. Total surface area = 24 + 60 + 50 + 50 = 184 cm².

Answer: 184 cm²

Cylinder

A cylinder has radius 4 cm and height 9 cm. Find its surface area in terms of π.

rhcurved face2 pi rheight hcylinder: 2 circles + curved rectanglesurface area = 2pi r2 + 2pi rh
Example: cylinder facesUse two circles plus the curved rectangle.
  1. Two circular ends: 2πr² = 2π x 4² = 32π.
  2. Curved face: 2πrh = 2π x 4 x 9 = 72π.
  3. Surface area = 32π + 72π = 104π cm².

Answer: 104π cm²

Common mistakes

  • Finding volume instead of surface area.
  • Using cubic units for surface area.
  • Forgetting that a cuboid has six faces.
  • Counting hidden joined faces in a compound solid when they are not on the outside.
  • For cylinders, using diameter instead of radius in πr².
  • For prisms, finding only the cross-section area and forgetting the side rectangles.

Quick exercise

Try these before moving to the exam-style questions.

  1. A cuboid is 4 cm by 5 cm by 6 cm. Find its surface area.
    lhlwlhlwwhwhsurface area means all outside facescuboid: 2lw + 2lh + 2wh
    Quick check: cuboidUse 2lw + 2lh + 2wh.
  2. A cube has side length 7 cm. Find its surface area.
    lhlwlhlwwhwhsurface area means all outside facescuboid: 2lw + 2lh + 2wh
    Quick check: cubeA cube has 6 equal square faces.
  3. A triangular prism has cross-section area 15 cm² and length 8 cm. What extra areas are still needed for surface area?
    trianglesame trianglerectanglessurface area of a prism2 x cross-section area + side rectangles
    Quick check: prism methodSurface area needs outside faces, not just the cross-section.
  4. A cylinder has radius 3 cm and height 10 cm. Find its curved surface area in terms of π.
    rhcurved face2 pi rheight hcylinder: 2 circles + curved rectanglesurface area = 2pi r2 + 2pi rh
    Quick check: curved faceCurved surface area is 2πrh.
  5. Surface area uses cm² or cm³?
    lhlwlhlwwhwhsurface area means all outside facescuboid: 2lw + 2lh + 2wh
    Quick check: unitsArea is always square units.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleFoundationNon-calculator3 marks

A cuboid has dimensions 10 cm, 4 cm and 3 cm. Work out its surface area.

lhlwlhlwwhwhsurface area means all outside facescuboid: 2lw + 2lh + 2wh
Question diagram: cuboidFind all six rectangular faces.
surface areacuboidfoundation geometry
Standard exam styleFoundation and HigherCalculator4 marks

A closed cylinder has radius 5 cm and height 12 cm. Work out the total surface area in terms of π.

rhcurved face2 pi rheight hcylinder: 2 circles + curved rectanglesurface area = 2pi r2 + 2pi rh
Question diagram: closed cylinderClosed cylinder means two circular ends.
surface areacylindercircle measures
ChallengeHigherCalculator5 marks

A triangular prism has triangular cross-section sides 5 cm, 12 cm and 13 cm. The perpendicular height to the 12 cm side is 5 cm. The prism length is 9 cm. Find the surface area.

trianglesame trianglerectanglessurface area of a prism2 x cross-section area + side rectangles
Question diagram: triangular prismAdd two triangular ends and three side rectangles.
surface areatriangular prismhigher geometry