Given an equilateral triangular prism with base edge 13 cm and height of 3 cm. Find the measure of the lateral area, surface area, and volume. Round to the nearest tenth. Given an equilateral triangular prism with base edge 13 cm and height of 3 cm. Find the measure of the lateral area, surface area, and volume. Round to the nearest tenth.
The Correct Answer and Explanation is :
To solve for the lateral area, surface area, and volume of the given equilateral triangular prism, we need to use specific formulas. The prism has a base edge of 13 cm and a height of 3 cm.
1. Lateral Area:
The lateral area of a prism is the area of the sides of the prism, excluding the bases. For a triangular prism, the lateral area is given by:
[
\text{Lateral Area} = \text{Perimeter of the base} \times \text{Height of the prism}
]
Step 1: Find the perimeter of the equilateral triangle base.
The perimeter (P) of an equilateral triangle with base edge (s) is:
[
P = 3 \times s = 3 \times 13 = 39 \, \text{cm}
]
Step 2: Calculate the lateral area.
Now, using the height (h = 3 \, \text{cm}), the lateral area is:
[
\text{Lateral Area} = 39 \times 3 = 117 \, \text{cm}^2
]
2. Surface Area:
The surface area of the prism is the sum of the lateral area and the area of the two triangular bases.
Step 1: Find the area of the triangular base.
The area (A_{\text{triangle}}) of an equilateral triangle is given by:
[
A_{\text{triangle}} = \frac{\sqrt{3}}{4} \times s^2
]
Substitute (s = 13):
[
A_{\text{triangle}} = \frac{\sqrt{3}}{4} \times 13^2 = \frac{\sqrt{3}}{4} \times 169 \approx 73.6 \, \text{cm}^2
]
Step 2: Calculate the surface area.
The surface area of the prism is:
[
\text{Surface Area} = \text{Lateral Area} + 2 \times A_{\text{triangle}} = 117 + 2 \times 73.6 = 117 + 147.2 = 264.2 \, \text{cm}^2
]
3. Volume:
The volume of the prism is the area of the triangular base multiplied by the height of the prism:
[
\text{Volume} = A_{\text{triangle}} \times h = 73.6 \times 3 = 220.8 \, \text{cm}^3
]
Final Answers:
- Lateral Area: (117 \, \text{cm}^2)
- Surface Area: (264.2 \, \text{cm}^2)
- Volume: (220.8 \, \text{cm}^3)