a pencil holder shaped like a triangular prism is shown. The height of the prism is 12 centimeters.
10 cm, 12 cm, 7 cm, 7 cm
What is the lateral surface area of the pencil holder in square centimeters?
The lateral surface area of a prism can be found by adding up the areas of all the faces that are not the base. In this case, the triangular bases are not included in the lateral surface area.
The triangular faces have side lengths of 10 cm, 12 cm, and 12 cm. Using Heron's formula, we can find the area of one of the triangular faces:
s = (a + b + c)/2 = (10 + 12 + 12)/2 = 17
Area = √(s(s-a)(s-b)(s-c)) = √(17(17-10)(17-12)(17-12)) = √(17*7*5*5) = √(17^2 * 5) = 17√5
There are two triangular faces, so the total area from the triangular faces is 2 * 17√5.
The rectangular faces have heights of 12 cm and lengths of 7 cm. The lateral surface area of each rectangular face is 12 * 7 = 84 cm². There are two rectangular faces, so the total area from the rectangular faces is 2 * 84 = 168 cm².
Therefore, the total lateral surface area of the pencil holder is 2 * 17√5 + 168 cm².