  # 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?

Answer: They are afraid that they will relax and unwind.

This is a 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 its height is 5 cm

To find the surface area of this square-based pyramid:

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 vertically to the base to form a right-angled triangle.

2. Apply Pythagoras' Theorem to find the height:  a² + b² = c²  but in this case, we need to find the length of b so we change the formula to:

c²- a² = b² where c = 9 and a = 3.5

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:

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

2.  Calculate the area of the triangle with the base of 7 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)

Example 3:

Find the area of this rectangular-based pyramid: 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 cm² = √53.76 cm² = 7.3 cm (correct to 1 decimal place)

Now we can find the 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) = 29.2 cm²

4.  Add the faces together: 20 + 18.25 + 18.25 + 14.6 + 14.6 = 85.7 cm² Where do Pharaohs like to eat?

Pizza Tut.

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