Question

2. The base of a right triangular pyramid has a side of 4 cm and the altitude of the pyramid is also 4 cm. Calculate the pyramid: (a) the base height AM; (b) the area of the base; (c) the length of the side edge SC.

Answer 1

Since the base of the pyramid is a triangle with a side length of 4 cm

That means the base is an equilateral triangle of sides 4 cm

The height of the equilateral triangle is

`h=(s)/(2)√(3)`

h is the height

s is the side

Since the side is 4 cm, then

s = 4

Substitute it in the rule above to find h

`\begin{gathered} h=(4)/(2)√(3) \\ \\ h=2√(3) \end{gathered}`

(a)

`AM=2√(3)`

Since the rule of the area of a triangle is

`A=(1)/(2)* b* h`

b is the base

h is the height

Since the base is 4, then

b = 4

Substitute the values of b and h in the rule of the area to find it

`\begin{gathered} A=(1)/(2)*4*2√(3) \\ \\ A=4√(3)\text{ cm}^2 \end{gathered}`

(b)

The area of the base is

`A=4√(3)\text{ cm}^2`

Since the altitude of the pyramid is 4 cm

We will use the Pythagoras theorem to find the side SC

From the triangle above we will use the height of the pyramid OS and a part of the height of the base = 2/3 h to find the side SC

`\begin{gathered} SC^2=4^2+((2)/(3)*2√(3))^2 \\ \\ SC^2=16+((4)/(3)√(3))^2 \\ \\ SC^2=16+(16)/(3) \\ \\ SC^2=(64)/(3) \end{gathered}`

Take a square root for both sides

`\begin{gathered} SC=\sqrt{(64)/(3)} \\ \\ SC=(8)/(√(3))*(√(3))/(√(3)) \\ \\ SC=(8)/(3)√(3) \end{gathered}`

(c)

The length of the side edge SC is

`SC=(8)/(3)√(3)\text{ cm}`