Question

Angel is decorating the outside of a box in the shape of a triangular pyramid. The figure below shows a net for the box. If all the triangles are equilateral, what is the surface area of the box, in square centimeters, that Angel decorates?

Answer 1

Solution

Step 1

Find the base area of the triangle of sides 12cm, by 12cm by 12cm using heron formula.

$\begin{gathered} \text{a = 12, b = 12, c = 12} \\ \text{s = }\frac{a+b+c}{2}\text{ = }\frac{12+12+12}{2}=\text{ }\frac{36}{2}\text{ = 18} \\ Area\text{ of the base = }\sqrt{s(s-a)(s-b)(s-c)} \\ \text{= }\sqrt{18(18-12)(18-12)(18-12)} \\ =\text{ }\sqrt{18\times6\times6\times6} \\ =\text{ 62.35} \end{gathered}$

Step 2

Find the area of the three triangular faces.

$\begin{gathered} Area\text{ of a triangle = }\frac{Base\text{ }\times\text{ Height}}{2} \\ =\text{ }\frac{12\text{ }\times10.39}{2} \\ =\text{ 62.34} \\ Area\text{ of the three faces = 3 }\times\text{ 62.34 = 187.02 cm}^2 \end{gathered}$

A = 62.35 + 187.02 = 249.37

Final answer

A = 249.36

Method 2