Question

a triangular pyramid and its net are shown. rain uses the calculations below to conclude that the surface area of the triangular pyramid is 238.5 square inches.
$3\left(\frac{1}{2} \times 10 \times 13\
ight) + \left(\frac{1}{2} \times 10 \times 8.7\
ight)$
$3(65) + 43.5$
$238.5$
is rain correct? use the drop - down menus to explain your reasoning.

Explanation:

Step1: Analyze the net of the triangular pyramid

A triangular pyramid (tetrahedron) with a triangular base has 4 triangular faces. From the net, the base is an isosceles triangle with base \(10\) in and height \(8.7\) in, and the three lateral faces are triangles with base \(10\) in and height \(12\) in (not \(13\) in as Rain used).

Step2: Calculate the area of lateral faces

The area of one lateral face is \(\frac{1}{2}\times10\times12 = 60\) square inches. There are 3 lateral faces, so total lateral surface area is \(3\times60 = 180\) square inches.

Step3: Calculate the area of the base

The area of the base (triangular face) is \(\frac{1}{2}\times10\times8.7=43.5\) square inches.

Step4: Calculate total surface area

Total surface area \(= 180 + 43.5=223.5\) square inches. Rain used the wrong height (\(13\) in instead of \(12\) in) for the lateral faces, so her calculation is incorrect.

Answer:

Rain is not correct. The lateral faces have a height of 12 inches (not 13 inches), so the area of each lateral face is \(\frac{1}{2}\times10\times12 = 60\), 3 lateral faces give \(3\times60 = 180\), plus the base area \(\frac{1}{2}\times10\times8.7 = 43.5\), total surface area is \(180 + 43.5 = 223.5\) square inches, not 238.5.