  # 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. 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

Why don't mummies go away on holiday?

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. Don't panic you have got this.

You will need to apply 3d trigonometry (Pythagoras' Theorem) to find the height. 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 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 cm&sup2; (x 2 for the opposite face) = 29.2cm&sup2;

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.

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