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.