Use the image to answer the question.
A triangular prism. The front triangular face is visible. The right rectangular face is visible. The sides and edges that are not visible are indicated by dashed lines. Parts are labeled with measurements. The left side of the triangle measures 8 centimeters. The left side meets the base to form a right angle. The base of the triangle measures 6 centimeters. The hypotenuse side of the triangle measures 10 centimeters. The width of the rectangular face measures 9 centimeters.
What is the surface area of the triangular prism?
To calculate the surface area of a triangular prism, you need to find the area of each face and then add them together.
1. The area of the front triangle can be found using the formula for the area of a triangle: 1/2 * base * height. Since it is a right triangle, the height can be found using the Pythagorean theorem: a^2 + b^2 = c^2. In this case, a = 6 cm, b = 8 cm, and c (the hypotenuse) = 10 cm. So, the height of the triangle is sqrt(10^2 - 6^2) = sqrt(100 - 36) = sqrt(64) = 8 cm. The area of the triangle is therefore 1/2 * 6 cm * 8 cm = 24 cm^2.
2. The area of the right rectangle is simply length * width, which is 8 cm * 9 cm = 72 cm^2.
3. Since the prism has two identical triangular faces (front and back) and three identical rectangular faces (top, bottom, and side), we can multiply the area of each face by 2 (for the triangles) or 3 (for the rectangles) to get the total surface area.
So, the total surface area of the triangular prism is 2(24 cm^2) + 3(72 cm^2) = 48 cm^2 + 216 cm^2 = 264 cm^2.