A pencil holder shaped like a triangular prism is shown. The height of the prism is 12 centimeters.
What is the lateral surface area of the pencil holder in square centimeters?
To find the lateral surface area of the triangular prism, we need to find the perimeter of the base and multiply it by the height.
The base is a triangle, and since it is a pencil holder, we can assume it is an equilateral triangle.
The lateral surface area = perimeter of the base x height.
Each side of the equilateral triangle is equal, so we need to find the length of one side of the triangle.
To find the length of one side, we can use the height of the prism as the height of an equilateral triangle.
Since an equilateral triangle has all sides equal, we can draw an altitude from one vertex to the base. This divides the equilateral triangle into two congruent 30-60-90 triangles. The side opposite the 60-degree angle is the height, which we know is 12 cm. We can use the ratios of the side lengths in a 30-60-90 triangle to find the length of one side.
In a 30-60-90 triangle, the ratio of the side lengths is 1: sqrt(3): 2.
Since the height of the triangle is opposite the 60-degree angle, which is the largest angle in the triangle, it corresponds to the longest side, which is the hypotenuse.
Using the ratios of a 30-60-90 triangle:
Length of one side = (sqrt(3)) * (height) = (sqrt(3)) * (12 cm) = 12(sqrt(3)) cm.
The perimeter of the base = 3 * (length of one side) = 3 * (12(sqrt(3)) cm) = 36(sqrt(3)) cm.
The lateral surface area = perimeter of the base x height = (36(sqrt(3)) cm) * (12 cm) = 432(sqrt(3)) cm^2.
Therefore, the lateral surface area of the pencil holder is 432(sqrt(3)) square centimeters.