Use the image to answer the question.

Question

Two front faces of a triangular pyramid are shown. Each face shows three levels of triangle shapes: 1 at the top, 3 in the middle row, and 5 along the bottom row. The orientation of the triangles alternates from upright to inverted on the bottom two rows. The sides of the triangular pyramid are equilateral and measure 4 inches. The vertical height measures 3.646 inches.

What is the surface area of Sally’s Pyraminx? Write your answer in decimal form. Round to the nearest thousandth.

(1 point)
in.2

Answer 1

The surface area of the triangular pyramid can be calculated by finding the area of each face and then adding them all together.

The area of an equilateral triangle can be calculated using the formula: A = (√3/4) * s^2, where s is the side length.

For the triangles on the top face:
A = (√3/4) * 4^2 = (√3/4) * 16 = 4√3 in^2

For the triangles on the middle row face:
A = 3 * [(√3/4) * 4^2] = 3 * 4√3 = 12√3 in^2

For the triangles on the bottom row face:
A = 5 * [(√3/4) * 4^2] = 5 * 4√3 = 20√3 in^2

Total surface area = 4√3 + 12√3 + 20√3 = 36√3 in^2

Using a calculator, the decimal approximation for the surface area is:
36√3 ≈ 62.225 in^2

Therefore, the surface area of Sally's Pyraminx is approximately 62.225 square inches.