Question
Total surface area of triangular prism with height of 4 base of 5 width of 6 and length of 12
Answers
Answered by
GPT-4o mini
To find the total surface area of a triangular prism, we need to calculate the area of the two triangular bases and the three rectangular faces.
1. **Calculate the area of the triangular base:**
The triangular base has a height of \( h = 4 \), a base of \( b = 5 \), and a width of \( w = 6 \) is not relevant here for a triangle because it suggests the triangle could be a right triangle. We will assume it's an isosceles triangle or right triangle since we need to clarify the dimensions used for a triangle.
If we treat the dimensions based simply on the height and base, the area \( A \) of the triangular base can be calculated as:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 4 = 10
\]
2. **Total area of the two triangular bases:**
\[
A_{bases} = 2 \times A_{triangle} = 2 \times 10 = 20
\]
3. **Calculate the area of the rectangular faces:**
The prism has three rectangular faces:
- The rectangle corresponding to the base of the triangle (5 units) and the length of the prism (12 units).
- The rectangle corresponding to the height of the triangle (4 units) and the length of the prism (12 units).
- The rectangle corresponding to the width of the triangle (6 units) and the length of the prism (12 units).
**Area of the first rectangular face:**
\[
A_{rect1} = \text{base} \times \text{length} = 5 \times 12 = 60
\]
**Area of the second rectangular face (height):**
\[
A_{rect2} = \text{height} \times \text{length} = 4 \times 12 = 48
\]
**Area of the third rectangular face (width):**
\[
A_{rect3} = \text{width} \times \text{length} = 6 \times 12 = 72
\]
**Total area of the rectangular faces:**
\[
A_{rectangles} = A_{rect1} + A_{rect2} + A_{rect3} = 60 + 48 + 72 = 180
\]
4. **Calculate the total surface area:**
\[
A_{total} = A_{bases} + A_{rectangles} = 20 + 180 = 200
\]
Therefore, the total surface area of the triangular prism is \( \boxed{200} \) square units.
1. **Calculate the area of the triangular base:**
The triangular base has a height of \( h = 4 \), a base of \( b = 5 \), and a width of \( w = 6 \) is not relevant here for a triangle because it suggests the triangle could be a right triangle. We will assume it's an isosceles triangle or right triangle since we need to clarify the dimensions used for a triangle.
If we treat the dimensions based simply on the height and base, the area \( A \) of the triangular base can be calculated as:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 4 = 10
\]
2. **Total area of the two triangular bases:**
\[
A_{bases} = 2 \times A_{triangle} = 2 \times 10 = 20
\]
3. **Calculate the area of the rectangular faces:**
The prism has three rectangular faces:
- The rectangle corresponding to the base of the triangle (5 units) and the length of the prism (12 units).
- The rectangle corresponding to the height of the triangle (4 units) and the length of the prism (12 units).
- The rectangle corresponding to the width of the triangle (6 units) and the length of the prism (12 units).
**Area of the first rectangular face:**
\[
A_{rect1} = \text{base} \times \text{length} = 5 \times 12 = 60
\]
**Area of the second rectangular face (height):**
\[
A_{rect2} = \text{height} \times \text{length} = 4 \times 12 = 48
\]
**Area of the third rectangular face (width):**
\[
A_{rect3} = \text{width} \times \text{length} = 6 \times 12 = 72
\]
**Total area of the rectangular faces:**
\[
A_{rectangles} = A_{rect1} + A_{rect2} + A_{rect3} = 60 + 48 + 72 = 180
\]
4. **Calculate the total surface area:**
\[
A_{total} = A_{bases} + A_{rectangles} = 20 + 180 = 200
\]
Therefore, the total surface area of the triangular prism is \( \boxed{200} \) square units.