Question

for a project, gabrielle made a model of an egyptian pyramid. the square base has a side length measuring 10 inches. the pyramid is 12 inches tall. what is the surface area of the model? 240 square inches 260 square inches 340 square inches 360 square inches

Explanation:

Step1: Calculate the slant height

Use the Pythagorean theorem \(l=\sqrt{h^{2}+(\frac{s}{2})^{2}}\), where \(h = 12\) inches (height of the pyramid) and \(s=10\) inches (side - length of the base).
\(l=\sqrt{12^{2}+(\frac{10}{2})^{2}}=\sqrt{144 + 25}=\sqrt{169}=13\) inches.

Step2: Calculate the area of the base

The area of the square base \(B=s^{2}\), with \(s = 10\) inches. So \(B=10\times10 = 100\) square inches.

Step3: Calculate the area of one triangular face

The area of a triangle \(A_{\triangle}=\frac{1}{2}\times base\times height\). Here, the base of the triangular face is \(s = 10\) inches and the height (slant height) \(l = 13\) inches. So \(A_{\triangle}=\frac{1}{2}\times10\times13=65\) square inches.

Step4: Calculate the total surface area

The total surface area of a square - based pyramid \(SA=B + 4A_{\triangle}\).
Substitute \(B = 100\) and \(A_{\triangle}=65\) into the formula: \(SA=100+4\times65=100 + 260=360\) square inches.

Answer:

360 square inches