Question
Find the total surface area of the figure. The base is 7 mm height is 4 mm. The length is 12 mm on the second area. It has a height of 5 mm and a width of 5 mm.
Answers
GPT-4o mini
To find the total surface area of the figure, we first need to clarify the shape based on the given dimensions. It sounds like it could be a combination of a rectangular prism (cuboid) and a triangular prism or some combination thereof. However, based on the information provided, let's assume you have a rectangular prism and a triangular prism.
### Rectangular Prism
Given:
- Length (l) = 12 mm
- Width (w) = 7 mm
- Height (h) = 4 mm
The formula for the surface area \(A\) of a rectangular prism is:
\[
A = 2(lw + lh + wh)
\]
Substituting the values:
- \(lw = 12 \times 7 = 84 \, \text{mm}^2\)
- \(lh = 12 \times 4 = 48 \, \text{mm}^2\)
- \(wh = 7 \times 4 = 28 \, \text{mm}^2\)
Now adding these up:
\[
A = 2(84 + 48 + 28) = 2(160) = 320 \, \text{mm}^2
\]
### Triangular Prism
For the second shape (assuming it's a triangular prism):
- Area of the triangular base = \(\frac{1}{2} \times \text{base} \times \text{height}\)
- Given base = 5 mm, height = 5 mm
The area of the triangle:
\[
\text{Area} = \frac{1}{2} \times 5 \times 5 = \frac{25}{2} = 12.5 \, \text{mm}^2
\]
The perimeter of the triangular base:
Assuming the triangular prism has equal sides (an equilateral triangle):
- Perimeter of base = \(5 + 5 + 5 = 15 \, \text{mm}\)
The lateral surface area of the triangular prism is given by:
\[
\text{Lateral Area} = \text{Perimeter} \times \text{Height}
\]
For height = 5 mm:
\[
\text{Lateral Area} = 15 \times 5 = 75 \, \text{mm}^2
\]
The total surface area of the triangular prism includes the lateral area plus the area of the two triangular bases:
\[
\text{Total Surface Area (Triangular Prism)} = \text{Lateral Area} + 2 \times \text{Area of triangle}
= 75 + 2 \times 12.5 = 75 + 25 = 100 \, \text{mm}^2
\]
### Total Surface Area
Finally, combining both shapes:
\[
\text{Total Surface Area} = \text{Surface Area of Rectangular Prism} + \text{Surface Area of Triangular Prism}
= 320 + 100 = 420 \, \text{mm}^2
\]
The total surface area of the figure is **420 mm²**.
### Rectangular Prism
Given:
- Length (l) = 12 mm
- Width (w) = 7 mm
- Height (h) = 4 mm
The formula for the surface area \(A\) of a rectangular prism is:
\[
A = 2(lw + lh + wh)
\]
Substituting the values:
- \(lw = 12 \times 7 = 84 \, \text{mm}^2\)
- \(lh = 12 \times 4 = 48 \, \text{mm}^2\)
- \(wh = 7 \times 4 = 28 \, \text{mm}^2\)
Now adding these up:
\[
A = 2(84 + 48 + 28) = 2(160) = 320 \, \text{mm}^2
\]
### Triangular Prism
For the second shape (assuming it's a triangular prism):
- Area of the triangular base = \(\frac{1}{2} \times \text{base} \times \text{height}\)
- Given base = 5 mm, height = 5 mm
The area of the triangle:
\[
\text{Area} = \frac{1}{2} \times 5 \times 5 = \frac{25}{2} = 12.5 \, \text{mm}^2
\]
The perimeter of the triangular base:
Assuming the triangular prism has equal sides (an equilateral triangle):
- Perimeter of base = \(5 + 5 + 5 = 15 \, \text{mm}\)
The lateral surface area of the triangular prism is given by:
\[
\text{Lateral Area} = \text{Perimeter} \times \text{Height}
\]
For height = 5 mm:
\[
\text{Lateral Area} = 15 \times 5 = 75 \, \text{mm}^2
\]
The total surface area of the triangular prism includes the lateral area plus the area of the two triangular bases:
\[
\text{Total Surface Area (Triangular Prism)} = \text{Lateral Area} + 2 \times \text{Area of triangle}
= 75 + 2 \times 12.5 = 75 + 25 = 100 \, \text{mm}^2
\]
### Total Surface Area
Finally, combining both shapes:
\[
\text{Total Surface Area} = \text{Surface Area of Rectangular Prism} + \text{Surface Area of Triangular Prism}
= 320 + 100 = 420 \, \text{mm}^2
\]
The total surface area of the figure is **420 mm²**.