What are the values of surface area and volume of a triangular prism, whose base is an equilateral triangle of side 6 inches and whose height is 1 foot?
To find the surface area and volume of a triangular prism, we need to know the dimensions of the base and the height of the prism.
Given:
Base: An equilateral triangle with side length 6 inches
Height: 1 foot (12 inches)
To calculate the surface area of the triangular prism, we need to find the area of each face and add them together. The triangular prism has two identical triangular faces and three rectangular faces.
1. Finding the area of the triangular faces:
The area of an equilateral triangle can be calculated using the formula: A = (√3 / 4) * side^2
Since the side length is 6 inches, the formula becomes: A = (√3 / 4) * 6^2
So the area of each triangular face is (√3 / 4) * 36 = (√3 / 4) * 36 = 9√3 square inches.
2. Finding the area of the rectangular faces:
The rectangular faces are created by the base and the height of the prism.
The base is an equilateral triangle, so each side length is 6 inches.
The height of the prism is 12 inches.
The area of a rectangle can be calculated using the formula: A = length * width
- Two rectangular faces have dimensions 6 inches (length) by 12 inches (height), so their areas are 6 * 12 = 72 square inches.
- One rectangular face has dimensions 6 inches (length) by 6 inches (width), so its area is 6 * 6 = 36 square inches.
3. Adding up the areas of the faces:
The surface area of the triangular prism is the sum of the areas of all five faces:
Total surface area = 2 * triangular face area + 3 * rectangular face area
= 2 * 9√3 + 3 * (72 + 36)
= 18√3 + 3 * 108
= 18√3 + 324 square inches
To calculate the volume of the triangular prism, we need to find the area of the base (equilateral triangle) and multiply it by the height.
1. Finding the area of the equilateral triangle:
The formula for the area of an equilateral triangle is the same as before:
A = (√3 / 4) * side^2
Substituting the side length of 6 inches, we have: A = (√3 / 4) * 6^2
The area of the equilateral triangle is (√3 / 4) * 36 = 9√3 square inches.
2. Calculating the volume:
The volume of any prism can be found by multiplying the area of the base by the height:
Volume = base area * height
= 9√3 * 12
= 108√3 cubic inches
Therefore, the surface area of the triangular prism is 18√3 + 324 square inches, and the volume is 108√3 cubic inches.