Revision NotesExam QuestionsPast Papers
O LevelMathsCambridgeRevision NotesLengths, Areas & VolumesVolume & Surface AreaSurface Area

Surface Area (Cambridge O Level Maths)

Revision Note

Amber

Author

Amber

Last updated

Did this video help you?

Surface Area

What is surface area?

  • A face is one of the flat or curved surfaces that make up a 3D shape
  • The surface area of a 3D shape is the sum of the areas of all the faces that make up the shape
  • Note how we are carrying a 2D idea (area) into 3 dimensions here

How do I find the surface area of cuboids, pyramids, and prisms?

  • In cuboids, polygonal-based pyramids, and polygonal-based prisms (ie. pyramids and prisms whose bases have straight sides), all the faces are flat
  • The surface area is found simply by adding up the areas of these flat faces
  • When calculating surface area, it can be very helpful to draw a 2D net for the 3D shape in question
  • For example:
    • The base of a square-based pyramid is 15 cm on a side
    • The triangular faces are identical isosceles triangles, each with a height (from the base to the top of the pyramid) of 23 cm
    • Find the total surface area of the pyramid
    • Draw a net for the shape:

Net-Square-Based-Pyramid, IGCSE & GCSE Maths revision notes

    • Area of square base =152 = 225 cm2
    • Area of one triangular face = ½ base × height = ½ × 15 × 23 =172.5 cm2
    • Total surface area =225 + 4 × 172.5 = 915 cm2

How do I find the surface area of cylinders, cones, and spheres?

  • All three of these shapes have curved faces, so we have to be a little more careful when calculating their surface areas

1. The net of a cylinder consists of two circles and a rectangle: Net-Cylinder, IGCSE & GCSE Maths revision notes

  • The curved surface area of a cylinder is bold 2 bold italic pi bold italic r bold italic h
  • The total surface area of a cylinder with base radius r and height h is therefore given by:

    Total surface area of a cylinder= bold 2 bold italic pi bold italic r to the power of bold 2 bold space bold plus bold space bold 2 bold italic pi bold italic r bold italic h

2. The net of a cone consists of the circular base along with the curved surface area:Net Cone, IGCSE & GCSE Maths revision notes 

  • The length l in that diagram is known as the slant height (while h is the vertical height of the cone)
  • To find the surface area of a cone with base radius r and slant height l, we use the formulae:
    • Curved surface area of a cone =bold italic pi bold italic r bold italic l
    • Total surface area of a cone =bold italic pi bold italic r to the power of bold 2 bold plus bold italic pi bold italic r bold italic l

3. To find the surface area of a sphere with radius r, use the formula:

  • Surface area of a sphere = bold 4 bold italic pi bold italic r to the power of bold 2

Sphere Radius r, IGCSE & GCSE Maths revision notes

  • Be careful when calculating the surface area of a hemisphere:
    The total surface area consists of the curved part (half of a sphere) PLUS the flat circular face – so the total surface area is 3πr2

Examiner Tip

  • The formula for the curved surface area of a cylinder and cone and the surface area of a sphere will be given to you on page 2 of the exam in the list of formulas
  • The rest of the formulae here come from what you should already know about areas of rectangles, triangles, and circles

Worked example

The base radius, r, of a cone is the same as the radius of a hemisphere. The total surface area of the cone is equal to the total surface area of the hemisphere. 

(a)
Find the slant height, l, of the cone in terms of r.
 
Find an expressions for the surface area of the hemisphere in terms of l and r.
Remember that a hemisphere has both a curved surface area and a flat circular face so the formula for the surface area is:
 
Surface area of hemisphere = 1 half cross times space 4 pi r squared space plus pi r squared space equals space 3 pi r squared
 
Find an expressions for the surface area of the cone in terms of l and r.
Remember that a cone has both a curved surface area and a flat circular face so the formula for the surface area is:
  
Surface area of cone = pi r l space plus pi r squared space equals space pi r open parentheses l space plus space r close parentheses
 
The surface areas are equal, so set these two formulae equal to each other.
 
3 pi r squared space equals space pi r open parentheses l space plus space r close parentheses
 
Rearrange to make l the subject.
Begin by dividing both sides by pi r.
 
table row cell 3 up diagonal strike pi r end strike squared space end cell equals cell space up diagonal strike pi r end strike open parentheses l space plus space r close parentheses end cell row cell 3 r space end cell equals cell space l space plus space r end cell end table
bold italic l bold space bold equals bold space bold 2 bold italic r
  
(b)
Given that r space equals space 19 cm, find the curved surface area of the cone.
GIve your answer accurate to 1 decimal place.
 
Use your answer from part (a) to find the value of l, by substituting r space equals space 19 into l space equals space 2 r.
 
l space equals space 2 r space equals space 2 space cross times space 19 space equals space 38 space cm
   
Substitute r space equals space 19 and l space equals space 38 into the formula for the curved surface area of the cone. 
Note that this is not for the whole surface area.
 
pi r l space equals space pi space cross times space 19 space cross times space 38 space equals space 722 straight pi space equals space 2268.229.... space
 
Round your answer to 1 decimal place. 
The first decimal place is a 2, and this is followed by a 2 so you do not need to round it up. 
 
bold Curved bold space bold surface bold space bold area bold space bold equals bold space bold 2268 bold. bold 2 bold space bold cm to the power of bold 2 bold space bold space stretchy left parenthesis 1 space d. p. stretchy right parenthesis

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourself

    Did this page help you?

    Previous:VolumeNext:Congruence & Similarity
    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.