The smart way to improve grades

Comprehensive & curriculum aligned

Affordable pricing from £10/month

Find the Surface area of a Pyramid

In this worksheet, students will learn the formula for finding the surface area of a pyramid involving 3D trigonometry.

'Find the Surface area of a Pyramid' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Geometry and Measures, Mensuration

Curriculum subtopic:   Mensuration and Calculation, Volume and Surface Area Calculations

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

ancient egyptian mummies

Question

Why don't mummies go away on holiday?

Answer

They are afraid that they will relax and unwind.

 

This is the clue to our topic.

We are going to look at the surface area of pyramids here with a bit of a twist.

Surface area is the area of each face of the pyramid added together.

So a quick recap.

 

Area of a square = base x height

Area of a rectangle = base x height

Area of a triangle = base x height ÷ 2

 

Example 1

 

This pyramid has a side length of 4 cm and it's height is 5 cm.

 

To find the surface area of this square based pyramid we

1. Calculate the area of the base  4 x 4 = 16 cm² 

2. Find the area of one of the triangular sides  4 x 5 ÷ 2 = 10 cm²

3.  There are four sides to the pyramid so 10 x 4 = 40 cm²

4. Add the total  area of the sides to the area of the base =  16 + 40 = 56 cm²

 

 

Example 2  - Here is the twist - there is no height given.

 

Find the surface area of this square based pyramid.

Pyramid no vertical height

Don't panic you have got this.

You will need to apply 3d trigonometry (Pythagoras' Theorem) to find the height.

3d pythagoras pyramid

The apex of the pyramid is exactly over the centre of the base. 

1. Draw a line in vertically to the base to form a right angled triangle.

2. Apply Pythagoras' Theorem to find the height.

9 cm² - 3.5 cm² =  √68.75 cm² = 8.3 cm (rounded to 1 decimal place)

Now you can find the surface area.

 

To find the area of this square based pyramid we

1. Calculate the area of the base  7 x 7 = 49 cm²

2.  We find the area of the triangle with the base of 5 cm, 7 x 8.3 ÷ 2 = 29.05 cm² 

3.  There are four triangles in total  29.05 x 4 = 116.2 cm²

4.  Add the base and the four triangles together 49 + 116.2  = 165.2 cm² (correct to 1 decimal place)

 

Find the area of this rectangular based pyramid.

Example 3

This is a little different so be careful

pyramid with 3d Pythagoras

Because this is rectangular we don't know whether to halve 4 cm or 5 cm. Instead we apply Pythagoras' Theorem twice to be accurate.

1. Find the length of the diagonal 5 cm² + 4 cm² = √41 cm² = 6.4 cm

2. Halve 6.4 to find the mid point 6.4 cm ÷ 2 = 3.2 cm

3. Calculate the height 8 cm² - 3.2 cm⊃2 = √53.76 cm² = 7.3 cm (correct to 1 decimal place.

Surface area

1. Base = 5 x 4 = 20 cm²

2. One triangle = 5 x 7.3 ÷ 2 = 18.25 cm² (x 2 for the opposite face) = 36.5 cm²

3. The other face triangle = 4 x 7.3 ÷ 2 = 14.6 cm² (x 2 for the opposite face) = 29.2cm²

4.  Add the faces together 20 + 18.25 + 18.25 + 14.6 + 14.6 = 85.7 cm²

 

 

Where do Pharaoh's like to eat?

Pizza Tut

Hopefully this has made you smile as you start the activities.

Find the surface area of this square based pyramid.

Round your answers to 1 decimal place throughout.

195.8 cm²

213.6 cm²

177.6 cm²

124.8 cm²

 

pyramid with no vertical height

What is the surface area of this pyramid?

Round your answer to 1 decimal place throughout.

Write your answer in figures without the units.

Find the surface area of this square based pyramid.

Side length 9 cm  and slant height 18 cm.

313.2 cm²

394.2 cm²

626.4 cm²

542.3 cm²

 

pyramid with no vertical height

Calculate the surface area of this square based pyramid.

Round your answers to 1 decimal place throughout.

Write your answer in figures without the units.

Find the surface area of this rectangular based pyramid.

rectangular based pyramid with no height

Write your answer in figures without the units.

Round your answers to 1 decimal place throughout.

 

 

 

pyramid no vertical height.

 

What is the surface area of this pyramid?

pyramid with no height

Round your answers to 1 decimal place as you work through.

 208 cm²198 cm²118 cm²98 cm²
Surface Area

                                            

Pyramid with no vertical height

What is the  surface area of this pyramid.

 

The surface area of this pyramid is 133.76 cm² 124.4 cm²

                                            

Find the surface area of this pyramid.

Round your answers to 1 decimal place throughout.

 566 cm²516 cm²536 cm²508.6cm²
Surface area

Find the surface area of this rectangular based pyramid.

Pyramid no vertical height

Round your answer to 1 decimal place throughout.

The surface area of this pyramid is 148.6 cm², 163.2 cm², 173.4 cm²
  • Question 1

Find the surface area of this square based pyramid.

Round your answers to 1 decimal place throughout.

CORRECT ANSWER
124.8 cm²
EDDIE SAYS
Did you remember to apply Pythagoras' Theorem first. This is the key to these types of question. Height 8 cm² - 3 cm² =√55 cm² = 7.4 cm Base area = 6 x 6 = 36 cm² Next find the area of 1 triangle 6 x 7.4 ÷ 2 = 22.2cm² Multiply the area of the triangle by 4. 22.2 x 4 = 88.8 cm² Now add this to the area of the base 88.8 + 36 = 124.8 cm² By the way, what music do mummies listen to? Rap
  • Question 2

 

pyramid with no vertical height

What is the surface area of this pyramid?

Round your answer to 1 decimal place throughout.

Write your answer in figures without the units.

CORRECT ANSWER
137.6
EDDIE SAYS
Let our friend Pythagoras' do his bit first. Height 10 cm² - 4 cm² =√84 cm² = 9.2 cm Base area = 8 x 8 = 64 cm² Next find the area of 1 triangle 8 x 9.2 ÷ 2 = 18.4 cm² 18.4 x 4 = 73.6 cm² 73.6 + 64 = 137.6 cm²
  • Question 3

Find the surface area of this square based pyramid.

Side length 9 cm  and slant height 18 cm.

CORRECT ANSWER
394.2 cm²
EDDIE SAYS
If in doubt sketch it out. Always. It really does help. Pythagoras' first. He never did like to be kept waiting. Height 18 cm² - 4.5 cm² =√303.75 cm² = 17.4cm Base area = 9 x 9 = 81 cm² Next find the area of 1 triangle 9 x 17.4 ÷ 2 = 78.3 cm² 78.3 x 4 = 313.2 cm² 313.2 + 81 = 394.2 cm²
  • Question 4

 

pyramid with no vertical height

Calculate the surface area of this square based pyramid.

Round your answers to 1 decimal place throughout.

Write your answer in figures without the units.

CORRECT ANSWER
68.6
EDDIE SAYS
Height 10 cm² - 1.5 cm² =√97.75 cm² = 9.9 cm Base area = 3 x 3 = 9 cm² Next find the area of 1 triangle 3 x 9.9 ÷ 2 = 14.9 cm² 14.9 x 4 = 59.6 cm² 59.6 + 9 = 68.6 cm² Brilliant stuff. This is a great way to show off more than one mathematical skill.
  • Question 5

Find the surface area of this rectangular based pyramid.

rectangular based pyramid with no height

Write your answer in figures without the units.

Round your answers to 1 decimal place throughout.

CORRECT ANSWER
116.6
EDDIE SAYS
Now its time to really show off. Here you have to apply Pythagoras theorem twice because we have a rectangular base. Find length of the diagonal first. 9 cm² + 7 cm² = √130 cm² = 11.4 cm 11.4 cm ÷ 2 = 5.7 cm Now we are ready to rock and roll. Height 8 cm² - 5.7 cm² =√31.51 cm² = 5.6 cm Base area = 9 x 7 = 63 cm² Next find the area of 1 triangle 9 x 5.6 ÷ 2 = 25.2 cm² Find the area of the other triangle 7 x 5.6 ÷ 2 = 19.6 cm² 63 + 25.2 + 25.2 + 19.6 + 19.6 = 116.6 cm²
  • Question 6

 

 

 

pyramid no vertical height.

CORRECT ANSWER
EDDIE SAYS
Find length of the diagonal first. 6 cm² + 4 cm² = √52 cm² = 7.2 cm 7.2 cm ÷ 2 =3.6 cm And we are off Height 9 cm² - 3.6 cm² =√68.04 cm² = 8.2cm Base area = 6 x 4 = 24 cm² Next find the area of 1 triangle 6 x 8.2 ÷ 2 = 24.6 cm² Find the area of the other triangle 4 x 8.2 ÷ 2 = 16.4 cm² 24 + 24.6 + 24.6 + 16.4 + 16.4 = 106 cm²
  • Question 7

 

What is the surface area of this pyramid?

pyramid with no height

Round your answers to 1 decimal place as you work through.

CORRECT ANSWER
 208 cm²198 cm²118 cm²98 cm²
Surface Area
EDDIE SAYS
Find length of the diagonal first. 8 cm² + 9 cm² = √145 cm² = 12.0 cm 12.0 cm ÷ 2 = 6 cm Here we go again. Height 10 cm² - 6 cm² =√64 cm² = 8 cm Base area = 8 x 9 = 72 cm² Next find the area of 1 triangle 8 x 8 ÷ 2 = 32 cm² Find the area of the other triangle 9 x 8 ÷ 2 = 36 cm² 72 + 32 + 32+ 36 + 36 = 208 cm²
  • Question 8

                                            

Pyramid with no vertical height

What is the  surface area of this pyramid.

 

CORRECT ANSWER
The surface area of this pyramid is 133.76 cm² 124.4 cm²
EDDIE SAYS
Find length of the diagonal first. 5 cm² + 4 cm² = √41 cm² = 6.4 cm 6.4 cm ÷ 2 = 3.2 cm Height 12 cm² - 3.2 cm² =√133.76 cm² = 11.6 cm Base area = 5 x 4 = 20 cm² Next find the area of 1 triangle 5 x 11.6 ÷ 2 = 29 cm² Find the area of the other triangle 4 x 11.6 ÷ 2 = 23.2 cm² 20+ 23.2 + 23.2 + 29 + 29 = 124.4 cm²
  • Question 9

                                            

Find the surface area of this pyramid.

Round your answers to 1 decimal place throughout.

CORRECT ANSWER
 566 cm²516 cm²536 cm²508.6cm²
Surface area
EDDIE SAYS
What did the Pharaoh say when he saw the pyramid? Mummies home. Are you home and dry with these yet? Find length of the diagonal first. 14 cm² + 12 cm² = √340 cm² = 18.4 cm 18.4 cm ÷ 2 = 9.2 cm Height 16 cm² - 9.2 cm² =√171.36 cm² = 13.1 cm Base area = 14 x 12 = 168 cm² Next find the area of 1 triangle 14 x 13.1 ÷ 2 = 91.7cm² Find the area of the other triangle 12 x 13.1 ÷ 2 = 78.6 cm² 168 + 91.7 + 91.7 + 78.6 + 78.6 = 508.6 cm²
  • Question 10

Find the surface area of this rectangular based pyramid.

Pyramid no vertical height

Round your answer to 1 decimal place throughout.

CORRECT ANSWER
The surface area of this pyramid is 148.6 cm², 163.2 cm², 173.4 cm²
EDDIE SAYS
And the final push. 9 cm² + 5 cm² = √106 cm² = 10.3 cm 10.3 cm ÷ 2 = 5.15cm And we are off Height 9 cm² - 5.15 cm² =√54.5 cm² = 7.4 cm Base area = 9 x 5 = 45 cm² Next find the area of 1 triangle 9 x 7.4 ÷ 2 = 33.3 cm² Find the area of the other triangle 5 x 7.4 ÷ 2 = 18.5 cm² 45 + 33.3 + 33.3+ 18.5 + 18.5 = 148.6 cm²
---- OR ----

Sign up for a £1 trial so you can track and measure your child's progress on this activity.

What is EdPlace?

We're your National Curriculum aligned online education content provider helping each child succeed in English, maths and science from year 1 to GCSE. With an EdPlace account you’ll be able to track and measure progress, helping each child achieve their best. We build confidence and attainment by personalising each child’s learning at a level that suits them.

Get started
laptop

Start your £1 trial today.
Subscribe from £10/month.