Calculate the surface area of a triangular prism shaped tent if the height is 9 ft., the length is 24 ft., the width is 20 ft., and the hypotenuse of the cover is 13.45
First, we need to find the area of the triangular ends of the prism. The base of the triangle is the width of the tent, which is 20 ft, and the height is the height of the prism, which is 9 ft. So, the area of each triangular end is:
1/2 x base x height = 1/2 x 20 ft x 9 ft = 90 sq ft
Next, we need to find the area of the rectangular sides of the prism. The length of the prism is 24 ft and the height is 9 ft, so the area of each side is:
length x height = 24 ft x 9 ft = 216 sq ft
Finally, we need to add up the areas of the two ends and the four sides to find the total surface area:
2 x 90 sq ft + 4 x 216 sq ft = 180 sq ft + 864 sq ft = 1044 sq ft
Therefore, the surface area of the tent is 1044 square feet.
To calculate the surface area of a triangular prism shaped tent, you need to calculate the area of each individual face and then add them together.
1. Calculate the area of the triangular base:
- The base of the triangle is given by the length and width of the tent, which are 24 ft and 20 ft respectively.
- The area of a triangle can be calculated using the formula: A = 0.5 * base * height.
- In this case, the base of the triangle is the width of the tent, so the area of the triangular base is A = 0.5 * 20 ft * 9 ft = 90 ft².
2. Calculate the area of the two rectangular sides:
- The two rectangular sides are identical in shape, so we only need to calculate the area of one side and double it.
- The dimensions of the rectangular side are given by the width and height of the tent, which are 20 ft and 9 ft respectively.
- The area of a rectangle can be calculated using the formula: A = length * width.
- In this case, the length of the rectangular side is the perimeter of the triangular base, which is equal to the sum of all three sides of the triangle.
- The hypotenuse of the triangular base is given as 13.45 ft. Since the triangular base is a right triangle, we can use the Pythagorean theorem to find the lengths of the other two sides.
- Let's denote the two unknown side lengths as a and b. We have a² + b² = c², where c is the hypotenuse.
- Substituting the given values, we have a² + b² = 13.45 ft².
- Solving for a and b, we find a ≈ 12.23 ft and b ≈ 5.14 ft.
- Therefore, the perimeter of the triangular base is P = a + b + c = 12.23 ft + 5.14 ft + 13.45 ft = 30.82 ft.
- Now we can calculate the area of one rectangular side: A = width * height = 20 ft * 9 ft = 180 ft².
- Finally, the total area of both rectangular sides is 2 * 180 ft² = 360 ft².
3. Calculate the total surface area:
- To find the total surface area, we need to sum up the area of the triangular base and the area of the two rectangular sides.
- Total Surface Area = Area of Triangular Base + 2 * Area of Rectangular Sides
- Total Surface Area = 90 ft² + 360 ft² = 450 ft².
Therefore, the surface area of the triangular prism shaped tent is 450 ft².