The edges of a tetrahedron (regular triangular pyramid) are each 4 cm . Leave the answers in surd form. a) Calculate the area of each face. b) Calculate the total surface area of the pyramid.

To solve the problem, we will first calculate the area of one face of the tetrahedron and then use that to find the total surface area. ### a) Area of each face A regular tetrahedron has triangular faces. The area \( A \) of an equilateral triangle with side length \( a \) can be calculated using the formula: \[ A = \frac{\sqrt{3}}{4} a^2 \] In this case, the side length \( a \) is 4 cm. Plugging in the value: \[ A = \frac{\sqrt{3}}{4} (4)^2 \] \[ A = \frac{\sqrt{3}}{4} \cdot 16 \] \[ A = 4\sqrt{3} \text{ cm}^2 \] So, the area of each face of the tetrahedron is \( 4\sqrt{3} \) cm². ### b) Total surface area of the pyramid A regular tetrahedron has 4 faces. Therefore, the total surface area \( S \) can be calculated as: \[ S = 4 \times \text{Area of one face} \] \[ S = 4 \times 4\sqrt{3} \] \[ S = 16\sqrt{3} \text{ cm}^2 \] Thus, the total surface area of the tetrahedron is \( 16\sqrt{3} \) cm². ### Summary of Answers - Area of each face: \( 4\sqrt{3} \) cm² - Total surface area: \( 16\sqrt{3} \) cm²