Question
The base of a triangular prism is 6. The height is 4. The length is 8. The width is 5. Find the lateral and surface area of the prism.
Answers
GPT-4o mini
To find the lateral area and surface area of a triangular prism, we need to determine the dimensions clearly and follow the appropriate formulas.
1. **Base Area of the Triangle:**
The base of the triangular prism is typically the base area of the triangular face. Here, you've provided the base (6) and the height (4) of the triangle.
\[
\text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 4 = 12
\]
2. **Lateral Area:**
The lateral surface area \( A_L \) of a prism can be found using the formula:
\[
A_L = \text{Perimeter of the base} \times \text{length}
\]
To find the perimeter of the triangular base, we need all three sides of the triangle. Given that the base is 6 and the height is 4, we can determine the lengths of the other sides using the Pythagorean theorem if we know that one of the vertices connects to the opposite height point, forming a right triangle.
Let's assume the triangle is a right triangle:
- One leg = 4 (height)
- Other leg = half the base = \( \frac{6}{2} = 3 \)
Therefore, the hypotenuse \( c \) can be calculated:
\[
c = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
So, the perimeter \( P \) of the triangle is:
\[
P = 6 + 4 + 5 = 15
\]
Now we calculate the lateral surface area:
\[
A_L = P \times \text{length} = 15 \times 8 = 120
\]
3. **Surface Area:**
The total surface area \( A_S \) of the prism is calculated by adding the lateral area and the areas of the two triangular bases:
\[
A_S = A_L + 2 \times \text{Base Area}
\]
Since we already found \( A_L = 120 \) and \( \text{Base Area} = 12 \):
\[
A_S = 120 + 2 \times 12 = 120 + 24 = 144
\]
Thus the final calculated areas are:
- **Lateral Area:** \( 120 \) square units
- **Surface Area:** \( 144 \) square units.
1. **Base Area of the Triangle:**
The base of the triangular prism is typically the base area of the triangular face. Here, you've provided the base (6) and the height (4) of the triangle.
\[
\text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 4 = 12
\]
2. **Lateral Area:**
The lateral surface area \( A_L \) of a prism can be found using the formula:
\[
A_L = \text{Perimeter of the base} \times \text{length}
\]
To find the perimeter of the triangular base, we need all three sides of the triangle. Given that the base is 6 and the height is 4, we can determine the lengths of the other sides using the Pythagorean theorem if we know that one of the vertices connects to the opposite height point, forming a right triangle.
Let's assume the triangle is a right triangle:
- One leg = 4 (height)
- Other leg = half the base = \( \frac{6}{2} = 3 \)
Therefore, the hypotenuse \( c \) can be calculated:
\[
c = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
So, the perimeter \( P \) of the triangle is:
\[
P = 6 + 4 + 5 = 15
\]
Now we calculate the lateral surface area:
\[
A_L = P \times \text{length} = 15 \times 8 = 120
\]
3. **Surface Area:**
The total surface area \( A_S \) of the prism is calculated by adding the lateral area and the areas of the two triangular bases:
\[
A_S = A_L + 2 \times \text{Base Area}
\]
Since we already found \( A_L = 120 \) and \( \text{Base Area} = 12 \):
\[
A_S = 120 + 2 \times 12 = 120 + 24 = 144
\]
Thus the final calculated areas are:
- **Lateral Area:** \( 120 \) square units
- **Surface Area:** \( 144 \) square units.