Question
or
A triangular prism is shown below.
Which diagram is a net for this prism?
What is the surface area of the triangular prism?
square centimeters
10 cm
6 cm
10 cm
16 cm
10 cm
Answers
Answered by
GPT-4o mini
To determine the net of a triangular prism, we need to consider its components. A triangular prism consists of two triangular bases and three rectangular lateral faces.
1. **Identify the bases:**
- The two bases are triangles with a base of 10 cm and a height of 6 cm.
2. **Identify the lateral faces:**
- The three rectangular faces will connect corresponding edges of the triangular bases. The dimensions of each rectangle depend on the lengths of the edges of the triangular base.
Let's denote the lengths:
- Base of the triangle (b) = 10 cm
- Height of the triangle (h) = 6 cm
- The other two sides of the triangle (let's call them s1 and s2) need to be determined. The question suggests dimensions of 10 cm and 16 cm, which could suggest two right triangles or other configurations based on the given dimensions.
3. **Calculating the Surface Area:**
- The surface area \(A\) of a triangular prism can be found using the formula:
\[
A = \text{Base Area} + \text{Lateral Area}
\]
- **Base Area** (Area of the triangle):
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \, \text{cm} \times 6 \, \text{cm} = 30 \, \text{cm}^2
\]
- Total base area for both triangles:
\[
\text{Total Base Area} = 2 \times 30 \, \text{cm}^2 = 60 \, \text{cm}^2
\]
- **Lateral Area** (Parallelograms corresponding to the rectangular faces):
- If we consider the heights to be the lengths of the three sides of the triangular face:
- Two sides 10 cm (for two rectangles) and one side which is the third dimension given (let's take 16 cm for the rectangular area parallel to the length):
\[
\text{Lateral Area} = (\text{Perimeter of triangle}) \times \text{Height of the prism}
\]
But, we need measurements for the other edges of the triangles. Without loss of generality, let’s use 10 and 16, and assuming both sides are equal for simplification:
- The perimeter of the triangles (assuming two equal sides = 10 cm):
\[
\text{Perimeter} = 10 \text{ (base)} + 10 + 16 = 36 \, \text{cm}
\]
- Now, calculating lateral area with a height of 10 cm (the height of the prism):
\[
\text{Lateral Area} = \text{Perimeter} \times \text{Height of the prism} = 36 \, \text{cm} \times 10 \, \text{cm} = 360 \, \text{cm}^2
\]
4. **Total Surface Area:**
- Total Surface Area:
\[
A = \text{Total Base Area} + \text{Lateral Area} = 60 \, \text{cm}^2 + 360 \, \text{cm}^2 = 420 \, \text{cm}^2
\]
So, the total surface area of the triangular prism is **420 square centimeters**.
If you need the description of the net which should contain all these shapes (2 triangles and proper rectangles), that should be checked against any specific diagrams provided in your material.
1. **Identify the bases:**
- The two bases are triangles with a base of 10 cm and a height of 6 cm.
2. **Identify the lateral faces:**
- The three rectangular faces will connect corresponding edges of the triangular bases. The dimensions of each rectangle depend on the lengths of the edges of the triangular base.
Let's denote the lengths:
- Base of the triangle (b) = 10 cm
- Height of the triangle (h) = 6 cm
- The other two sides of the triangle (let's call them s1 and s2) need to be determined. The question suggests dimensions of 10 cm and 16 cm, which could suggest two right triangles or other configurations based on the given dimensions.
3. **Calculating the Surface Area:**
- The surface area \(A\) of a triangular prism can be found using the formula:
\[
A = \text{Base Area} + \text{Lateral Area}
\]
- **Base Area** (Area of the triangle):
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \, \text{cm} \times 6 \, \text{cm} = 30 \, \text{cm}^2
\]
- Total base area for both triangles:
\[
\text{Total Base Area} = 2 \times 30 \, \text{cm}^2 = 60 \, \text{cm}^2
\]
- **Lateral Area** (Parallelograms corresponding to the rectangular faces):
- If we consider the heights to be the lengths of the three sides of the triangular face:
- Two sides 10 cm (for two rectangles) and one side which is the third dimension given (let's take 16 cm for the rectangular area parallel to the length):
\[
\text{Lateral Area} = (\text{Perimeter of triangle}) \times \text{Height of the prism}
\]
But, we need measurements for the other edges of the triangles. Without loss of generality, let’s use 10 and 16, and assuming both sides are equal for simplification:
- The perimeter of the triangles (assuming two equal sides = 10 cm):
\[
\text{Perimeter} = 10 \text{ (base)} + 10 + 16 = 36 \, \text{cm}
\]
- Now, calculating lateral area with a height of 10 cm (the height of the prism):
\[
\text{Lateral Area} = \text{Perimeter} \times \text{Height of the prism} = 36 \, \text{cm} \times 10 \, \text{cm} = 360 \, \text{cm}^2
\]
4. **Total Surface Area:**
- Total Surface Area:
\[
A = \text{Total Base Area} + \text{Lateral Area} = 60 \, \text{cm}^2 + 360 \, \text{cm}^2 = 420 \, \text{cm}^2
\]
So, the total surface area of the triangular prism is **420 square centimeters**.
If you need the description of the net which should contain all these shapes (2 triangles and proper rectangles), that should be checked against any specific diagrams provided in your material.