Question
Find the TOTAL surface area of this triangular prism in cm^2.
B = 6cm, H = 4cm, L = 12cm, and two 5cm but I don't know where they go.
1. 24cm^2
2. 192cm^2
3. 202cm^2
4. 216cm^2
Then find the LATERAL surface area of this triangular prism in cm^2.
1. 192cm^2
2. 216cm^2
3. 45cm^2
4. 76cm^2
B = 6cm, H = 4cm, L = 12cm, and two 5cm but I don't know where they go.
1. 24cm^2
2. 192cm^2
3. 202cm^2
4. 216cm^2
Then find the LATERAL surface area of this triangular prism in cm^2.
1. 192cm^2
2. 216cm^2
3. 45cm^2
4. 76cm^2
Answers
GPT-4o mini
To find the total surface area and lateral surface area of the triangular prism, we need to understand the parameters given:
- Base \( B = 6 \, \text{cm} \)
- Height \( H = 4 \, \text{cm} \)
- Length \( L = 12 \, \text{cm} \)
### Step 1: Calculate the area of the triangular base
The area \( A \) of a triangle can be calculated with the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the given values for the base and height of the triangle:
\[
A = \frac{1}{2} \times 6 \, \text{cm} \times 4 \, \text{cm} = \frac{1}{2} \times 24 \, \text{cm}^2 = 12 \, \text{cm}^2
\]
### Step 2: Calculate the total surface area
A triangular prism has two triangular bases and three rectangular faces.
1. **Area of the triangular bases**:
There are two triangular bases, so:
\[
\text{Area of two triangular bases} = 2 \times A = 2 \times 12 \, \text{cm}^2 = 24 \, \text{cm}^2
\]
2. **Area of the rectangular faces**:
- Two sides of the triangle (6 cm and the height base not identified) can be calculated using the Pythagorean theorem to find the third side (slant sides):
\[
\text{Slant side} = \sqrt{(B/2)^2 + H^2} = \sqrt{(6/2)^2 + 4^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{cm}
\]
- So, the three edges of the rectangular faces are:
- 1 rectangle of \( 6 \times 12 \)
- 1 rectangle of \( 5 \times 12 \) (two of these)
- Calculate each rectangle's area:
\[
\text{Area of first rectangle} = 6 \, \text{cm} \times 12 \, \text{cm} = 72 \, \text{cm}^2
\]
\[
\text{Area of second rectangle} = 5 \, \text{cm} \times 12 \, \text{cm} = 60 \, \text{cm}^2 \quad (\text{this area occurs twice})
\]
\[
\text{2 of second rectangles' area} = 2 \times 60 \, \text{cm}^2 = 120 \, \text{cm}^2
\]
3. **Total lateral surface area**:
\[
\text{Total lateral surface area} = 72 \, \text{cm}^2 + 120 \, \text{cm}^2 = 192 \, \text{cm}^2
\]
### Step 3: Calculate total surface area
Now, combining the base area and lateral area:
\[
\text{Total Surface Area} = \text{Area of bases} + \text{Lateral Area} = 24 \, \text{cm}^2 + 192 \, \text{cm}^2 = 216 \, \text{cm}^2
\]
### Final Answers
1. **Total surface area**: **216 cm²**.
2. **Lateral surface area**: **192 cm²**.
**Total Surface Area:** 4. 216 cm²
**Lateral Surface Area:** 1. 192 cm²
- Base \( B = 6 \, \text{cm} \)
- Height \( H = 4 \, \text{cm} \)
- Length \( L = 12 \, \text{cm} \)
### Step 1: Calculate the area of the triangular base
The area \( A \) of a triangle can be calculated with the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the given values for the base and height of the triangle:
\[
A = \frac{1}{2} \times 6 \, \text{cm} \times 4 \, \text{cm} = \frac{1}{2} \times 24 \, \text{cm}^2 = 12 \, \text{cm}^2
\]
### Step 2: Calculate the total surface area
A triangular prism has two triangular bases and three rectangular faces.
1. **Area of the triangular bases**:
There are two triangular bases, so:
\[
\text{Area of two triangular bases} = 2 \times A = 2 \times 12 \, \text{cm}^2 = 24 \, \text{cm}^2
\]
2. **Area of the rectangular faces**:
- Two sides of the triangle (6 cm and the height base not identified) can be calculated using the Pythagorean theorem to find the third side (slant sides):
\[
\text{Slant side} = \sqrt{(B/2)^2 + H^2} = \sqrt{(6/2)^2 + 4^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{cm}
\]
- So, the three edges of the rectangular faces are:
- 1 rectangle of \( 6 \times 12 \)
- 1 rectangle of \( 5 \times 12 \) (two of these)
- Calculate each rectangle's area:
\[
\text{Area of first rectangle} = 6 \, \text{cm} \times 12 \, \text{cm} = 72 \, \text{cm}^2
\]
\[
\text{Area of second rectangle} = 5 \, \text{cm} \times 12 \, \text{cm} = 60 \, \text{cm}^2 \quad (\text{this area occurs twice})
\]
\[
\text{2 of second rectangles' area} = 2 \times 60 \, \text{cm}^2 = 120 \, \text{cm}^2
\]
3. **Total lateral surface area**:
\[
\text{Total lateral surface area} = 72 \, \text{cm}^2 + 120 \, \text{cm}^2 = 192 \, \text{cm}^2
\]
### Step 3: Calculate total surface area
Now, combining the base area and lateral area:
\[
\text{Total Surface Area} = \text{Area of bases} + \text{Lateral Area} = 24 \, \text{cm}^2 + 192 \, \text{cm}^2 = 216 \, \text{cm}^2
\]
### Final Answers
1. **Total surface area**: **216 cm²**.
2. **Lateral surface area**: **192 cm²**.
**Total Surface Area:** 4. 216 cm²
**Lateral Surface Area:** 1. 192 cm²