The bases of a rectangular prism are squares whose perimeters are 36 inches. If the height of the prism is 8 inches, what is the surface are of the prism?
The surface area of a prism consists of two parts:
1. area of the bases
2. area of the sides.
Even without knowing the dimensions of the bases, the area of the sides can be calculated if the perimeter is known:
Side area
= perimeter of base * height
= 36 inches * 8 inches
= 288 square inches.
Since the area of the bases are not known, you can only calculate the area of the sides, which is probably the answer required.
To find the surface area of the rectangular prism, we need to find the areas of the six faces and add them together.
Step 1: Calculate the perimeter of one base.
The perimeter of a square is given by the formula P = 4s, where s is the length of one side. Since the perimeter of the base is 36 inches, each side of the square base has a length of 36/4 = 9 inches.
Step 2: Calculate the area of one base.
The area of a square is given by the formula A = s^2, where s is the length of one side. Using the length calculated in step 1, the area of one base is 9 inches^2 = 81 square inches.
Step 3: Calculate the area of the other base.
Since the prism is rectangular, the second base has the same area as the first base, which is 81 square inches.
Step 4: Calculate the area of the other four faces.
The height of the prism is given as 8 inches. The other four faces of the prism are rectangles, each with a length of 9 inches (the side of the square base) and a height of 8 inches.
To calculate the area of each rectangle: A = length x height = 9 inches x 8 inches = 72 square inches.
Since there are four of these faces, the total area of these four rectangles is 4 x 72 square inches = 288 square inches.
Step 5: Add up the areas of all six faces.
The surface area of the prism is the sum of the areas of all six faces:
Surface area = Area of Base 1 + Area of Base 2 + Area of Four Faces
Surface area = 81 square inches + 81 square inches + 288 square inches
Surface area = 450 square inches
Therefore, the surface area of the rectangular prism is 450 square inches.
To find out the surface area of the prism, we need to calculate the area of each face and then sum them up.
First, let's find the length of each side of the square base:
- The perimeter of a square is given by 4 times the length of one side.
- Since the perimeter of the square base is 36 inches, the length of one side is 36 / 4 = 9 inches.
Now, let's calculate the area of each face:
- The area of one of the square bases is length * width = 9 inches * 9 inches = 81 square inches.
- The prism has two square bases, so the total area of the bases is 2 * 81 square inches = 162 square inches.
Next, let's calculate the area of the remaining four rectangular faces:
- Each rectangular face has a length equal to the length of the side of the square base, which is 9 inches.
- The width of each rectangular face is equal to the height of the prism, which is 8 inches.
- So, the area of each rectangular face is 9 inches * 8 inches = 72 square inches.
- There are four rectangular faces, so the total area of the four faces is 4 * 72 square inches = 288 square inches.
Finally, let's calculate the total surface area of the prism:
- The total surface area is the sum of the areas of all the faces.
- It is given by 2 times the area of the bases plus the area of the four rectangular faces.
- So, the total surface area = 2 (162 square inches) + 288 square inches = 612 square inches.
Therefore, the surface area of the rectangular prism is 612 square inches.