Question
The net of an equilateral triangular prism and some of it’s dimensions are shown in the diagram the height of each triangular base is 4.5 cm what is the lateral area of the equilateral triangular prism
Answers
GPT-4o mini
To find the lateral area of an equilateral triangular prism, we need to determine the dimensions of the triangular base and the height (length) of the prism.
1. **Understanding the Triangular Base:**
An equilateral triangle has all sides equal and all angles equal to \(60^\circ\). Since we are given that the height of each triangular base is 4.5 cm, we can use the properties of an equilateral triangle to find the side length.
The height \(h\) of an equilateral triangle with side length \(s\) is given by the formula:
\[
h = \frac{\sqrt{3}}{2} s
\]
Rearranging this to solve for \(s\):
\[
s = \frac{2h}{\sqrt{3}}
\]
Substituting \(h = 4.5\) cm:
\[
s = \frac{2 \times 4.5}{\sqrt{3}} = \frac{9}{\sqrt{3}} = 3\sqrt{3} \, \text{cm}
\]
2. **Lateral Area of the Prism:**
The lateral area \(A_L\) of a prism is given by the formula:
\[
A_L = \text{Perimeter of the base} \times \text{Height of the prism}
\]
The perimeter \(P\) of the equilateral triangle is:
\[
P = 3s = 3(3\sqrt{3}) = 9\sqrt{3} \, \text{cm}
\]
Let \(h_p\) be the height (length) of the prism. If it’s not given, we cannot calculate the lateral area exactly without this value. Assuming the height \(h_p\) of the prism is \(h_p\) cm, we would calculate:
\[
A_L = P \times h_p = (9\sqrt{3}) \times h_p
\]
If you have a specific height for the prism, just replace \(h_p\) with that value to get the lateral area. If \(h_p\) is given, please provide it or use this formula to compute the lateral area of the equilateral triangular prism.
1. **Understanding the Triangular Base:**
An equilateral triangle has all sides equal and all angles equal to \(60^\circ\). Since we are given that the height of each triangular base is 4.5 cm, we can use the properties of an equilateral triangle to find the side length.
The height \(h\) of an equilateral triangle with side length \(s\) is given by the formula:
\[
h = \frac{\sqrt{3}}{2} s
\]
Rearranging this to solve for \(s\):
\[
s = \frac{2h}{\sqrt{3}}
\]
Substituting \(h = 4.5\) cm:
\[
s = \frac{2 \times 4.5}{\sqrt{3}} = \frac{9}{\sqrt{3}} = 3\sqrt{3} \, \text{cm}
\]
2. **Lateral Area of the Prism:**
The lateral area \(A_L\) of a prism is given by the formula:
\[
A_L = \text{Perimeter of the base} \times \text{Height of the prism}
\]
The perimeter \(P\) of the equilateral triangle is:
\[
P = 3s = 3(3\sqrt{3}) = 9\sqrt{3} \, \text{cm}
\]
Let \(h_p\) be the height (length) of the prism. If it’s not given, we cannot calculate the lateral area exactly without this value. Assuming the height \(h_p\) of the prism is \(h_p\) cm, we would calculate:
\[
A_L = P \times h_p = (9\sqrt{3}) \times h_p
\]
If you have a specific height for the prism, just replace \(h_p\) with that value to get the lateral area. If \(h_p\) is given, please provide it or use this formula to compute the lateral area of the equilateral triangular prism.