Question

1. a triangular pyramid has faces that are all equilateral triangles. each side length is 16 cm. what is the surface area of the pyramid?

Explanation:

Step1: Find the area of one equilateral triangle

The formula for the area of an equilateral triangle with side length \(a\) is \(A=\frac{\sqrt{3}}{4}a^{2}\). Given \(a = 16\mathrm{cm}\), then \(A=\frac{\sqrt{3}}{4}\times16^{2}=\frac{\sqrt{3}}{4}\times256 = 64\sqrt{3}\mathrm{cm}^{2}\)

Step2: Calculate the surface area of the pyramid

A triangular pyramid (tetrahedron) with all - face equilateral triangles has \(4\) faces. The surface area \(S\) of the pyramid is \(S = 4A\). Substitute \(A=64\sqrt{3}\mathrm{cm}^{2}\) into the formula, we get \(S=4\times64\sqrt{3}=256\sqrt{3}\approx256\times1.732 = 443.392\mathrm{cm}^{2}\)

Answer:

The surface area of the pyramid is \(256\sqrt{3}\mathrm{cm}^{2}\approx443.4\mathrm{cm}^{2}\)