EdPlace's GCSE home learning maths lesson: Surface Area

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Get them started on the lesson below and then jump into our teacher-created activities to practice what they've learnt. We've recommended five to ensure they feel secure in their knowledge - 5-a-day helps keeps the learning loss at bay (or so we think!).

Are they keen to start practising straight away? Head to the bottom of the page to find the activities. 

Now...onto the lesson!


GCSE Specification Requirements 
GCSE students should be able to calculate the areas of circles and composite shapes; surface area and volume of spheres, pyramids, cones and composite shapes.

We're Scratching More Than The Surface With This Topic!

You spent hours choosing the perfect present for your child.  You wrapped it up carefully and were so excited to give it to them on their birthday and …… drum roll…… they were more interested in the box it came in.  How often did that happen when they were younger?  Little did you know, that at a young age they were taking an interest in maths and surface area!  Now that they're that bit older they need to be able to demonstrate an understanding of this childhood pastime.  

We're confident that if you follow the step-by-step approach below your child will be able to:

1) Recall the different formulae required to find the surface area of any shape

2) Apply this knowledge to solve problems

3) Explain their reasoning clearly


Finding the Right Formula

To get started we need to know the formulae for finding the area of certain shapes. It's important to check that your child knows them as only the more complex ones are given on an exam paper. These formulae are just something your child will have to learn.  Why not try this first part as a quiz to assess your child’s current understanding. They might surprise you with what they know!

Area of a square = base x height

Area of a rectangle = base x height

Area of a parallelogram = base x height (nothing like a bit of repetition).

Area of a triangle = base x height ÷ 2 (or ½ base x height)

Note:  The height is the vertical height of the triangle, not the slant height.

Area of a trapezium = top + bottom ÷ 2. Then multiply by the height. (Vertical height).

Area of a circle  =πr² sometimes written as x radius x radius.

Top Tip – to learn formula connected with a circle think of pies.  Yes, that’s right, pies! (Read on for our clever mnemonics!)
π is a number that never ends...3.14159265 is the start of it, so sometimes you will see mathematicians shorten it to 3.142!  Why not challenge your child to learn it to 20 numbers?

Circumference = π x diameter   (Cherry Pies are Delicious)

Area = π r²    (Apple Pies aRToo)

Once your child knows these, all the hard work is done and makes what looks to be tricky on the 'surface' nothing more than a walk in the park.  There are other surface area formulae that you child will need to be familiar with, like, the area of a cylinder, but these types of formulae are usually given on an examination paper.


Step 1 - Using the (not so) Secret Formula

Next check that your child understands what area is and can apply the correct formula.  Area is the measurement of a given surface and is always measured in units squared Therefore, the area of any 2D shape is the amount of space inside the boundary lines.  Read on to see the formulae in action, you'll notice that we always include the correct units.

The area of a square with 4 cm sides is 4 x 4 =16 cm²

The area of a triangle of base 8 cm and vertical height 7 cm is 8 x 7 = 56 cm ÷ 2 = 28 cm².   A common mistake here is that students forget to divide by 2

The area of a trapezium with the top length of 7 cm, the bottom length of 13 cm and a height of 8 cm   7 + 13 = 20       20 ÷ 2 = 10        10 x 8 = 80 cm²

The area of a circle with a radius of 4 cm π (or 3.14) x 4 x 4 = 50.24 cm²

That is all very well, I hear you ask but how can I explain surface area?


Step 2 - Seeing in 3D...

Now the fun begins!  You need to apply the formulae learnt above to find the area of 3D shapes.  As with any maths question encourage your child to break it down into smaller chunks.  Always write down the answers as you go, students often try to retain all the information in their head until the last minute and this is where mistakes are made! Let’s look at the surface area of a cube.  As you work through this problem with your child ask them what it is they already know.  Hopefully, they can come up with the following:

1) Define area

2) They can see that a cube has all square faces

3) Bonus point is that all the faces will be the same size.

4) The area of a square is base x height

The surface area of any shape is the area of each face worked out and then added together.  You can demonstrate this really easily if you have a dice to hand.  For example, a cube has a side of 5 cm.  First, calculate the area of 1 face  5 x 5 = 25 cm².  Secondly, as a cube has 6 sides that are all equal multiply 25 cm² x 6 = 150 cm².

Note: This calculation not only shows a teacher and examiner that your child knows how to calculate the area of the basic shape but also shows how this is used to find the surface area.  Using the correct unit of measurement puts the icing on the cake.  Simply writing the answer doesn't show off knowledge and reduces marks in exams.  


Step 3 - Starting to Take Shape!

In this next example we'll be using the formula that we're now confident with but this time we're working out the surface area of a rectangular prism.  Again, if you want to really bring this to life look around for any kind of box in your house!

Hopefully, we can see what we need to do here, we know the formula for working out the area of a 2D rectangle, the only thing that is quite different is that this time all of the faces (surfaces) are not the same length.  So, we need to consider this when we calculate the surface area.

The area of the square face = 3 x 3 = 9 cm²  and there are 2 of these faces = 9 cm² x 2 = 18 cm² 

The area of the rectangular face = 3 x 6 = 18 cm²  and there are 4 of these faces = 18 cm² x 4 = 72 cm² 

Finally, we add this together and we know that the surface area of this rectangular prism  = 18 cm² + 72 cm² = 90 cm² 


This next question requires the same approach.  We need to consider the number of faces and how they differ from one another in their area, before adding these figures altogether.

Triangular Prism

There are 5 faces in this triangular prism.

Let's start with the square face at the bottom 3 cm x 3 cm = 9 cm² 

Let's now calulare the slanting face area 3 cm x 5 cm = 15 cm²  

The upright rectangular face we know is  3 cm x 4 cm = 12 cm²  

The triangular surface is (3 cm x 4 cm) ÷ 2 = 6 cm² and we know there are two of these faces so 6 cm² x 2 = 12 cm²

Now, we add these numbers up and get 9 + 15 + 12 + 12 = 48 cm² 


Step 4 - Surface Area in Practice!

Before moving on to more complex problems it is always worth consolidating what you've covered with your child.

1) What is the formula for finding the area of a triangle?

2) What is the formula for finding the area of a circle?

3) What is the formula for finding the area of a trapezium?

4) What is meant by surface area?

5) What is the value of π (Pi)?


Step 5 - Give it a go...

Check your child’s progress and enhance their knowledge of surface area with the following activities.  

The links below will guide you through the surface area of more complex shapes.

All activities are created by teachers and automatically marked. Plus, with an EdPlace subscription, we can automatically progress your child at a level that's right for them. Sending you progress reports along the way so you can track and measure progress, together - brilliant! 

Activity 1 - Surface Area of a Cuboid

Activity 2 - Surface Area of Cylinders

Activity 3 - Find the Surface Area of a Pyramid

Activity 4 - Surface Area of a Cone

Activity 5 - Surface Area of a Sphere



1) Base x height ÷ 2

2) π x radius x radius (or πr2)

3) Top + bottom ÷ 2 x height

4) The area of every face of a 3D shape added together.

5) 3.14


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All English, maths and science from Year 1 - GCSE