The figure below is a net for a triangular prism. Side a = 46 inches, side b = 10 inches, side c = 31 inches, and altitude d = 21 inches. What is the surface area of this figure?
We can start by breaking down the figure into its three rectangular faces and two triangular faces.
The rectangular faces have dimensions 46 x 21, 10 x 21, and 31 x 21.
The triangular faces have base 46, 10, and 31, with a height of 21.
To find the area of a triangle with base b and height h, we use the formula A = (1/2)bh.
So the area of the triangular faces are:
- (1/2)(46)(21) = 483 square inches
- (1/2)(10)(21) = 105 square inches
- (1/2)(31)(21) = 325.5 square inches
To find the area of a rectangle with length l and width w, we use the formula A = lw.
So the area of the rectangular faces are:
- 46 x 21 = 966 square inches
- 10 x 21 = 210 square inches
- 31 x 21 = 651 square inches
Adding up all the areas, we get:
483 + 105 + 325.5 + 966 + 210 + 651 = 2740.5
Therefore, the surface area of the triangular prism is approximately 2740.5 square inches.
To find the surface area of the triangular prism, you need to calculate the areas of the individual faces and add them together.
1. Base Triangle:
- The base triangle has sides a, b, and c.
- Use Heron's formula to calculate its area.
- Heron's formula states: Area = √(s(s-a)(s-b)(s-c)), where s is the semiperimeter.
- The semiperimeter (s) is given by s = (a + b + c)/2.
Let's calculate the semiperimeter and area of the base triangle:
s = (a + b + c)/2
= (46 + 10 + 31)/2
= 87/2
= 43.5
Area = √(s(s-a)(s-b)(s-c))
= √(43.5(43.5-46)(43.5-10)(43.5-31))
= √(43.5(-2.5)(33.5)(12.5))
= √(43.5 * -2.5 * 33.5 * 12.5)
≈ √(36440.625)
≈ 190.91 square inches (rounded to two decimal places)
2. Two Lateral Faces:
- Each lateral face is a rectangle.
- The length of each lateral face is equal to the perimeter of the base triangle, which is (a + b + c) = 46 + 10 + 31 = 87 inches.
- The height (altitude) of the prism is given as d = 21 inches.
Let's calculate the areas of the two lateral faces:
Area of one lateral face = length * height
= 87 * 21
= 1827 square inches
Since there are two lateral faces, the total area of the lateral faces is 2 * 1827 = 3654 square inches.
3. Total Surface Area:
- The total surface area of the prism is the sum of the base triangle area and the lateral faces' areas.
Total Surface Area = Area of the base triangle + Total area of the lateral faces
= 190.91 + 3654
≈ 3844.91 square inches
Therefore, the surface area of the given triangular prism is approximately 3844.91 square inches.