<---replaces #3--->The figure below is a net for a triangular prism. Side a = 22 feet, side b = 12 feet, side c = 16 feet, side d = 16 feet, and side e = 20 feet. What is the surface area of this figure?
We can start by breaking the net into its individual faces.
There are 3 rectangular faces:
- One face has dimensions 22 feet by 12 feet
- Another face has dimensions 16 feet by 12 feet
- The third face has dimensions 22 feet by 16 feet
There are 2 triangular faces:
- One face has base 16 feet and height 20 feet
- The other face has base 22 feet and height 20 feet
To find the surface area, we need to add up the areas of all five faces.
The area of each rectangular face is (length x width).
- First face: 22 ft x 12 ft = 264 sq ft
- Second face: 16 ft x 12 ft = 192 sq ft
- Third face: 22 ft x 16 ft = 352 sq ft
The area of each triangular face is (1/2 x base x height).
- First triangular face: (1/2) x 16 ft x 20 ft = 160 sq ft
- Second triangular face: (1/2) x 22 ft x 20 ft = 220 sq ft
Now we add up the areas:
264 + 192 + 352 + 160 + 220 = 1188 sq ft
Therefore, the surface area of this triangular prism is 1188 square feet.
To find the surface area of the triangular prism, we need to calculate the areas of all the individual faces and then sum them up.
1. Start by identifying the different faces of the triangular prism:
- Base 1: Triangle formed by sides a, b, and c.
- Base 2: Triangle formed by sides d, e, and f.
- Lateral Face 1: Rectangle formed by sides a and d.
- Lateral Face 2: Rectangle formed by sides b and e.
- Lateral Face 3: Rectangle formed by sides c and f.
2. Calculate the area of Base 1 (triangle):
- Use Heron's formula to find the area of the triangle.
- Semi-perimeter, s = (a + b + c)/2 = (22 + 12 + 16)/2 = 25
- Area of Base 1, A₁ = √(s(s - a)(s - b)(s - c))
3. Calculate the area of Base 2 (triangle):
- Use Heron's formula to find the area of the triangle.
- Semi-perimeter, s = (d + e + f)/2 = (16 + 20 + 16)/2 = 26
- Area of Base 2, A₂ = √(s(s - d)(s - e)(s - f))
4. Calculate the area of Lateral Face 1 (rectangle):
- Area = length × width
- Lateral Face 1, A₃ = a × d
5. Calculate the area of Lateral Face 2 (rectangle):
- Area = length × width
- Lateral Face 2, A₄ = b × e
6. Calculate the area of Lateral Face 3 (rectangle):
- Area = length × width
- Lateral Face 3, A₅ = c × f
7. Add up the areas of all the faces to find the surface area:
- Surface Area = A₁ + A₂ + A₃ + A₄ + A₅
Now, let's calculate the values and find the surface area of the triangular prism.