Question

The side walls of a regular quadrilateral pyramid are equilateral triangles with sides equal to 8 cm. Calculate the pyramid:1) the sum of the lengths of all the sides;2) the area of the base;3) the length of the height of the side wall;4) the area of the side wall.

Answer 1

a) 64cm

b) 64 square cm

c) 4√3 cm

d) 16√3 square cm

Explanations:

A regular quadrilateral pyramid with equilateral triangles is as shown below;

1) The pyramid has 8 side lengths, hence the sum of the length of all the sides is given as:

`\begin{gathered} Sum\text{ of side lengths}=8*8cm \\ Sum\text{ of side lengths}=64cm \end{gathered}`

2) Since the triangular sides are equilateral, the base of the pyramid will be a square with side length of 8cm. The area of the base is expressed as:

`\begin{gathered} A=length* length \\ A=8cm*8cm \\ A=64cm^2 \end{gathered}`

3) Since one side wall is an equilateral triangle, the height will be perpendicular to the base as shown:

In order to determine the height, we will use the Pythagorean theorem as shown:

`\begin{gathered} 8^2=h^2+4^2 \\ h^2=8^2-4^2 \\ h^2=64-16 \\ h^2=48 \\ h=√(48)=4√(3)cm \\ \end{gathered}`

4) The area of the side wall is equivalent to the area of the triangle expressed as:

`\begin{gathered} Area\text{ of side wall}=(1)/(2)* base* height \\ Area\text{ of side wall}=(1)/(2)*8cm*4√(3) \\ Area\text{ of side wall}=16√(3)cm^2 \end{gathered}`