Question

use the net for the triangular pyramid to find the surface area.

Explanation:

Step1: Calculate area of base triangle

The formula for the area of a triangle is \(A=\frac{1}{2}\times base\times height\). For the base triangle with base \(b = 6\space cm\) and height \(h=6\space cm\), \(A_{base}=\frac{1}{2}\times6\times6 = 18\space cm^{2}\).

Step2: Calculate area of one lateral triangle

For a lateral triangle with base \(b = 6\space cm\) and height \(h = 9\space cm\), \(A_{lateral1}=\frac{1}{2}\times6\times9=27\space cm^{2}\).

Step3: Calculate area of another lateral triangle

For the other lateral triangle with base \(b = 6\space cm\) and height \(h = 3\space cm\), \(A_{lateral2}=\frac{1}{2}\times6\times3 = 9\space cm^{2}\).

Step4: Find total surface area

The net of the triangular pyramid has \(1\) base and \(3\) lateral faces. The surface area \(S=A_{base}+ 2\times A_{lateral1}+A_{lateral2}\). Substitute the values: \(S = 18+2\times27 + 9\). First, \(2\times27=54\). Then \(18 + 54+9=81\space cm^{2}\).

Answer:

\(81\space cm^{2}\)