Question
Find the surface area of a right rectangular pyramid with these measurements l = 8 cm, w = 4 cm, and h = 2 cm. Exspress your answer as a decimal rounded to the nearest huindreth
Answers
Answer
its wrong.. :(
Answer
its wrong....
Answer
itsw rong and now ic ant change my answer..
Answered by
GPT-4o mini
To find the surface area of a right rectangular pyramid, we need to calculate the area of the base and the area of the triangular faces that make up the sides of the pyramid.
1. **Calculate the area of the base:**
The base of the pyramid is rectangular, with length \( l = 8 \) cm and width \( w = 4 \) cm.
\[
\text{Area of the base} = l \times w = 8 \, \text{cm} \times 4 \, \text{cm} = 32 \, \text{cm}^2
\]
2. **Calculate the slant heights of the triangular faces:**
The height of the pyramid is \( h = 2 \) cm. We need to find the slant heights of the triangular faces that connect the apex of the pyramid to the edges of the base.
- For the triangular face with base length \( l = 8 \) cm:
- The half-length of the base is \( \frac{l}{2} = 4 \, \text{cm} \).
- We can use the Pythagorean theorem to find the slant height \( l_1 \):
\[
l_1 = \sqrt{ \left( \frac{l}{2} \right)^2 + h^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \, \text{cm}
\]
- For the triangular face with base width \( w = 4 \) cm:
- The half-width of the base is \( \frac{w}{2} = 2 \, \text{cm} \).
- We can find the slant height \( l_2 \):
\[
l_2 = \sqrt{ \left( \frac{w}{2} \right)^2 + h^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \, \text{cm}
\]
3. **Calculate the area of the triangular faces:**
- There are two triangular faces with base \( l \):
\[
\text{Area of 2 triangular faces} = 2 \cdot \left( \frac{1}{2} \times l \times l_1 \right) = 2 \cdot \left( \frac{1}{2} \times 8 \cdot 2\sqrt{5} \right) = 8 \cdot 2\sqrt{5} = 16\sqrt{5} \, \text{cm}^2
\]
- There are also two triangular faces with base \( w \):
\[
\text{Area of 2 triangular faces} = 2 \cdot \left( \frac{1}{2} \times w \times l_2 \right) = 2 \cdot \left( \frac{1}{2} \times 4 \cdot 2\sqrt{2} \right) = 4 \cdot 2\sqrt{2} = 8\sqrt{2} \, \text{cm}^2
\]
4. **Combine the areas to find the total surface area:**
\[
\text{Total surface area} = \text{Area of the base} + \text{Area of the triangular faces}
\]
\[
\text{Total surface area} = 32 + 16\sqrt{5} + 8\sqrt{2}
\]
5. **Calculate numerical values for the square roots:**
- Approximate \(\sqrt{5} \approx 2.236\)
- Approximate \(\sqrt{2} \approx 1.414\)
Thus,
\[
16\sqrt{5} \approx 16 \times 2.236 \approx 35.776\, \text{cm}^2
\]
\[
8\sqrt{2} \approx 8 \times 1.414 \approx 11.312\, \text{cm}^2
\]
Adding these values:
\[
\text{Total surface area} \approx 32 + 35.776 + 11.312 \approx 79.088 \, \text{cm}^2
\]
Rounding to the nearest hundredth:
\[
\text{Total surface area} \approx 79.09 \, \text{cm}^2
\]
Thus, the total surface area of the right rectangular pyramid is approximately **79.09 cm²**.
1. **Calculate the area of the base:**
The base of the pyramid is rectangular, with length \( l = 8 \) cm and width \( w = 4 \) cm.
\[
\text{Area of the base} = l \times w = 8 \, \text{cm} \times 4 \, \text{cm} = 32 \, \text{cm}^2
\]
2. **Calculate the slant heights of the triangular faces:**
The height of the pyramid is \( h = 2 \) cm. We need to find the slant heights of the triangular faces that connect the apex of the pyramid to the edges of the base.
- For the triangular face with base length \( l = 8 \) cm:
- The half-length of the base is \( \frac{l}{2} = 4 \, \text{cm} \).
- We can use the Pythagorean theorem to find the slant height \( l_1 \):
\[
l_1 = \sqrt{ \left( \frac{l}{2} \right)^2 + h^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \, \text{cm}
\]
- For the triangular face with base width \( w = 4 \) cm:
- The half-width of the base is \( \frac{w}{2} = 2 \, \text{cm} \).
- We can find the slant height \( l_2 \):
\[
l_2 = \sqrt{ \left( \frac{w}{2} \right)^2 + h^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \, \text{cm}
\]
3. **Calculate the area of the triangular faces:**
- There are two triangular faces with base \( l \):
\[
\text{Area of 2 triangular faces} = 2 \cdot \left( \frac{1}{2} \times l \times l_1 \right) = 2 \cdot \left( \frac{1}{2} \times 8 \cdot 2\sqrt{5} \right) = 8 \cdot 2\sqrt{5} = 16\sqrt{5} \, \text{cm}^2
\]
- There are also two triangular faces with base \( w \):
\[
\text{Area of 2 triangular faces} = 2 \cdot \left( \frac{1}{2} \times w \times l_2 \right) = 2 \cdot \left( \frac{1}{2} \times 4 \cdot 2\sqrt{2} \right) = 4 \cdot 2\sqrt{2} = 8\sqrt{2} \, \text{cm}^2
\]
4. **Combine the areas to find the total surface area:**
\[
\text{Total surface area} = \text{Area of the base} + \text{Area of the triangular faces}
\]
\[
\text{Total surface area} = 32 + 16\sqrt{5} + 8\sqrt{2}
\]
5. **Calculate numerical values for the square roots:**
- Approximate \(\sqrt{5} \approx 2.236\)
- Approximate \(\sqrt{2} \approx 1.414\)
Thus,
\[
16\sqrt{5} \approx 16 \times 2.236 \approx 35.776\, \text{cm}^2
\]
\[
8\sqrt{2} \approx 8 \times 1.414 \approx 11.312\, \text{cm}^2
\]
Adding these values:
\[
\text{Total surface area} \approx 32 + 35.776 + 11.312 \approx 79.088 \, \text{cm}^2
\]
Rounding to the nearest hundredth:
\[
\text{Total surface area} \approx 79.09 \, \text{cm}^2
\]
Thus, the total surface area of the right rectangular pyramid is approximately **79.09 cm²**.
Answered by
GPT-4o mini
Let’s reevaluate the calculation for the surface area of the right rectangular pyramid step by step to ensure accuracy.
### Given Data
- Length of the base \( l = 8 \, \text{cm} \)
- Width of the base \( w = 4 \, \text{cm} \)
- Height of the pyramid \( h = 2 \, \text{cm} \)
### Steps to Find the Surface Area
1. **Calculate the Area of the Base**
\[
\text{Area of the base} = l \times w = 8 \times 4 = 32 \, \text{cm}^2
\]
2. **Calculate the Slant Heights of the Triangular Faces**
- **Slant height \( l_1 \)** for the triangular face with base \( l = 8 \, \text{cm} \):
- The half-length of the base is \( \frac{l}{2} = 4 \, \text{cm} \).
\[
l_1 = \sqrt{ \left( \frac{l}{2} \right)^2 + h^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \, \text{cm}
\]
- **Slant height \( l_2 \)** for the triangular face with base \( w = 4 \, \text{cm} \):
- The half-width of the base is \( \frac{w}{2} = 2 \, \text{cm} \).
\[
l_2 = \sqrt{ \left( \frac{w}{2} \right)^2 + h^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \, \text{cm}
\]
3. **Calculate the Area of the Triangular Faces**
- **For the triangular faces with base \( l \)**:
\[
\text{Area of 2 triangular faces} = 2 \cdot \left( \frac{1}{2} \times l \times l_1 \right) = 2 \cdot \left( \frac{1}{2} \times 8 \times 2\sqrt{5} \right) = 2 \cdot (8\sqrt{5}) = 16\sqrt{5} \, \text{cm}^2
\]
- **For the triangular faces with base \( w \)**:
\[
\text{Area of 2 triangular faces} = 2 \cdot \left( \frac{1}{2} \times w \times l_2 \right) = 2 \cdot \left( \frac{1}{2} \times 4 \times 2\sqrt{2} \right) = 2 \cdot (4\sqrt{2}) = 8\sqrt{2} \, \text{cm}^2
\]
4. **Combine the Areas for Total Surface Area**
\[
\text{Total Surface Area} = \text{Area of base} + \text{Area of triangular faces}
\]
\[
\text{Total Surface Area} = 32 + 16\sqrt{5} + 8\sqrt{2}
\]
5. **Calculate Numerical Values for Final Answer**
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt{2} \approx 1.414 \)
Substituting these into the area calculations:
\[
16\sqrt{5} \approx 16 \times 2.236 \approx 35.776 \, \text{cm}^2
\]
\[
8\sqrt{2} \approx 8 \times 1.414 \approx 11.312 \, \text{cm}^2
\]
Now combine the areas:
\[
\text{Total Surface Area} \approx 32 + 35.776 + 11.312 \approx 79.088 \, \text{cm}^2
\]
### Rounding
So we round this to:
\[
\text{Total Surface Area} \approx 79.09 \, \text{cm}^2
\]
This calculation follows the same logic, and it appears the value is consistent. If there’s a specific step or calculation that you think is incorrect, please let me know, and I will clarify.
### Given Data
- Length of the base \( l = 8 \, \text{cm} \)
- Width of the base \( w = 4 \, \text{cm} \)
- Height of the pyramid \( h = 2 \, \text{cm} \)
### Steps to Find the Surface Area
1. **Calculate the Area of the Base**
\[
\text{Area of the base} = l \times w = 8 \times 4 = 32 \, \text{cm}^2
\]
2. **Calculate the Slant Heights of the Triangular Faces**
- **Slant height \( l_1 \)** for the triangular face with base \( l = 8 \, \text{cm} \):
- The half-length of the base is \( \frac{l}{2} = 4 \, \text{cm} \).
\[
l_1 = \sqrt{ \left( \frac{l}{2} \right)^2 + h^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \, \text{cm}
\]
- **Slant height \( l_2 \)** for the triangular face with base \( w = 4 \, \text{cm} \):
- The half-width of the base is \( \frac{w}{2} = 2 \, \text{cm} \).
\[
l_2 = \sqrt{ \left( \frac{w}{2} \right)^2 + h^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \, \text{cm}
\]
3. **Calculate the Area of the Triangular Faces**
- **For the triangular faces with base \( l \)**:
\[
\text{Area of 2 triangular faces} = 2 \cdot \left( \frac{1}{2} \times l \times l_1 \right) = 2 \cdot \left( \frac{1}{2} \times 8 \times 2\sqrt{5} \right) = 2 \cdot (8\sqrt{5}) = 16\sqrt{5} \, \text{cm}^2
\]
- **For the triangular faces with base \( w \)**:
\[
\text{Area of 2 triangular faces} = 2 \cdot \left( \frac{1}{2} \times w \times l_2 \right) = 2 \cdot \left( \frac{1}{2} \times 4 \times 2\sqrt{2} \right) = 2 \cdot (4\sqrt{2}) = 8\sqrt{2} \, \text{cm}^2
\]
4. **Combine the Areas for Total Surface Area**
\[
\text{Total Surface Area} = \text{Area of base} + \text{Area of triangular faces}
\]
\[
\text{Total Surface Area} = 32 + 16\sqrt{5} + 8\sqrt{2}
\]
5. **Calculate Numerical Values for Final Answer**
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt{2} \approx 1.414 \)
Substituting these into the area calculations:
\[
16\sqrt{5} \approx 16 \times 2.236 \approx 35.776 \, \text{cm}^2
\]
\[
8\sqrt{2} \approx 8 \times 1.414 \approx 11.312 \, \text{cm}^2
\]
Now combine the areas:
\[
\text{Total Surface Area} \approx 32 + 35.776 + 11.312 \approx 79.088 \, \text{cm}^2
\]
### Rounding
So we round this to:
\[
\text{Total Surface Area} \approx 79.09 \, \text{cm}^2
\]
This calculation follows the same logic, and it appears the value is consistent. If there’s a specific step or calculation that you think is incorrect, please let me know, and I will clarify.
Answered by
GPT-4o mini
I apologize for the confusion caused by my previous calculations. Let’s go through the process of calculating the surface area of a right rectangular pyramid more carefully.
### Given:
- Length \( l = 8 \, \text{cm} \)
- Width \( w = 4 \, \text{cm} \)
- Height \( h = 2 \, \text{cm} \)
### Steps to Find the Surface Area:
1. **Calculate the Area of the Base:**
The base of the pyramid is a rectangle.
\[
\text{Area of the base} = l \times w = 8 \, \text{cm} \times 4 \, \text{cm} = 32 \, \text{cm}^2
\]
2. **Calculate the Slant Heights of the Triangular Faces:**
- **For the triangular face with base \( l = 8 \, \text{cm} \):**
- Half the length of the base:
\[
\frac{l}{2} = \frac{8}{2} = 4 \, \text{cm}
\]
- The slant height \( l_1 \):
\[
l_1 = \sqrt{ \left( \frac{l}{2} \right)^2 + h^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \, \text{cm}
\]
- **For the triangular face with base \( w = 4 \, \text{cm} \):**
- Half the width of the base:
\[
\frac{w}{2} = \frac{4}{2} = 2 \, \text{cm}
\]
- The slant height \( l_2 \):
\[
l_2 = \sqrt{ \left( \frac{w}{2} \right)^2 + h^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \, \text{cm}
\]
3. **Calculate the Area of the Triangular Faces:**
- **Two triangular faces with base \( l = 8 \, \text{cm} \):**
\[
\text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 2\sqrt{5} = 8\sqrt{5} \, \text{cm}^2
\]
- Total area for the two triangular faces:
\[
2 \times 8\sqrt{5} = 16\sqrt{5} \, \text{cm}^2
\]
- **Two triangular faces with base \( w = 4 \, \text{cm} \):**
\[
\text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 2\sqrt{2} = 4\sqrt{2} \, \text{cm}^2
\]
- Total area for the two triangular faces:
\[
2 \times 4\sqrt{2} = 8\sqrt{2} \, \text{cm}^2
\]
4. **Combine the Areas to Find the Total Surface Area:**
\[
\text{Total Surface Area} = \text{Area of Base} + \text{Area of Triangular Faces}
\]
\[
\text{Total Surface Area} = 32\, \text{cm}^2 + 16 \sqrt{5} \, \text{cm}^2 + 8 \sqrt{2} \, \text{cm}^2
\]
5. **Calculating Numerical Values for the Square Roots:**
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt{2} \approx 1.414 \)
Now substitute these values back:
- For \( 16 \sqrt{5} \):
\[
16 \sqrt{5} \approx 16 \times 2.236 \approx 35.776 \, \text{cm}^2
\]
- For \( 8 \sqrt{2} \):
\[
8 \sqrt{2} \approx 8 \times 1.414 \approx 11.312 \, \text{cm}^2
\]
6. **Adding Everything Together:**
\[
\text{Total Surface Area} \approx 32 + 35.776 + 11.312 \approx 79.088 \, \text{cm}^2
\]
### Final Answer:
Thus, rounding to the nearest hundredth, the total surface area is approximately:
\[
\text{Total Surface Area} \approx 79.09 \, \text{cm}^2
\]
Please confirm if this matches with your expectations, or if there's any specific point that seems incorrect. Thank you for your patience!
### Given:
- Length \( l = 8 \, \text{cm} \)
- Width \( w = 4 \, \text{cm} \)
- Height \( h = 2 \, \text{cm} \)
### Steps to Find the Surface Area:
1. **Calculate the Area of the Base:**
The base of the pyramid is a rectangle.
\[
\text{Area of the base} = l \times w = 8 \, \text{cm} \times 4 \, \text{cm} = 32 \, \text{cm}^2
\]
2. **Calculate the Slant Heights of the Triangular Faces:**
- **For the triangular face with base \( l = 8 \, \text{cm} \):**
- Half the length of the base:
\[
\frac{l}{2} = \frac{8}{2} = 4 \, \text{cm}
\]
- The slant height \( l_1 \):
\[
l_1 = \sqrt{ \left( \frac{l}{2} \right)^2 + h^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \, \text{cm}
\]
- **For the triangular face with base \( w = 4 \, \text{cm} \):**
- Half the width of the base:
\[
\frac{w}{2} = \frac{4}{2} = 2 \, \text{cm}
\]
- The slant height \( l_2 \):
\[
l_2 = \sqrt{ \left( \frac{w}{2} \right)^2 + h^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \, \text{cm}
\]
3. **Calculate the Area of the Triangular Faces:**
- **Two triangular faces with base \( l = 8 \, \text{cm} \):**
\[
\text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 2\sqrt{5} = 8\sqrt{5} \, \text{cm}^2
\]
- Total area for the two triangular faces:
\[
2 \times 8\sqrt{5} = 16\sqrt{5} \, \text{cm}^2
\]
- **Two triangular faces with base \( w = 4 \, \text{cm} \):**
\[
\text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 2\sqrt{2} = 4\sqrt{2} \, \text{cm}^2
\]
- Total area for the two triangular faces:
\[
2 \times 4\sqrt{2} = 8\sqrt{2} \, \text{cm}^2
\]
4. **Combine the Areas to Find the Total Surface Area:**
\[
\text{Total Surface Area} = \text{Area of Base} + \text{Area of Triangular Faces}
\]
\[
\text{Total Surface Area} = 32\, \text{cm}^2 + 16 \sqrt{5} \, \text{cm}^2 + 8 \sqrt{2} \, \text{cm}^2
\]
5. **Calculating Numerical Values for the Square Roots:**
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt{2} \approx 1.414 \)
Now substitute these values back:
- For \( 16 \sqrt{5} \):
\[
16 \sqrt{5} \approx 16 \times 2.236 \approx 35.776 \, \text{cm}^2
\]
- For \( 8 \sqrt{2} \):
\[
8 \sqrt{2} \approx 8 \times 1.414 \approx 11.312 \, \text{cm}^2
\]
6. **Adding Everything Together:**
\[
\text{Total Surface Area} \approx 32 + 35.776 + 11.312 \approx 79.088 \, \text{cm}^2
\]
### Final Answer:
Thus, rounding to the nearest hundredth, the total surface area is approximately:
\[
\text{Total Surface Area} \approx 79.09 \, \text{cm}^2
\]
Please confirm if this matches with your expectations, or if there's any specific point that seems incorrect. Thank you for your patience!
Answered by
GPT-4o mini
I sincerely apologize for the confusion and any distress caused by the errors in my previous calculations. Let’s approach this again from scratch, ensuring clarity and accuracy.
To find the surface area \( A \) of a right rectangular pyramid, we need to do the following steps:
### 1. Surface Area Formula
The surface area of a right rectangular pyramid consists of the area of the base plus the areas of the triangular faces. The formula can be represented as:
\[
A = \text{Base Area} + \text{Lateral Area}
\]
### 2. Given Measurements
- Length of the base \( l = 8 \, \text{cm} \)
- Width of the base \( w = 4 \, \text{cm} \)
- Height of the pyramid \( h = 2 \, \text{cm} \)
### 3. Calculate the Base Area
The area of the rectangular base is:
\[
\text{Base Area} = l \times w = 8 \times 4 = 32 \, \text{cm}^2
\]
### 4. Calculate the Slant Heights
To find the slant heights of the triangular faces, we will need to determine the height of each triangular face using the Pythagorean theorem.
- **For triangular faces that have a base of length \( l \) (8 cm)**:
- Half of the base length is:
\[
\frac{l}{2} = \frac{8}{2} = 4 \, \text{cm}
\]
- The slant height \( h_1 \) can be calculated as:
\[
h_1 = \sqrt{\left(\frac{l}{2}\right)^2 + h^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \, \text{cm}
\]
- **For triangular faces that have a base of width \( w \) (4 cm)**:
- Half of the base width is:
\[
\frac{w}{2} = \frac{4}{2} = 2 \, \text{cm}
\]
- The slant height \( h_2 \) can be calculated as:
\[
h_2 = \sqrt{\left(\frac{w}{2}\right)^2 + h^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \, \text{cm}
\]
### 5. Calculate the Areas of the Triangular Faces
Now, calculate the areas of the triangular faces.
- **Area of the two triangles with base \( l \) (8 cm)**:
\[
\text{Area}_{l} = 2 \cdot \left(\frac{1}{2} \times l \times h_1\right) = 2 \cdot \left(\frac{1}{2} \times 8 \times 2\sqrt{5}\right)
\]
\[
= 2 \cdot (8\sqrt{5}) = 16\sqrt{5} \, \text{cm}^2
\]
- **Area of the two triangles with base \( w \) (4 cm)**:
\[
\text{Area}_{w} = 2 \cdot \left(\frac{1}{2} \times w \times h_2\right) = 2 \cdot \left(\frac{1}{2} \times 4 \times 2\sqrt{2}\right)
\]
\[
= 2 \cdot (4\sqrt{2}) = 8\sqrt{2} \, \text{cm}^2
\]
### 6. Calculate the Total Surface Area
Now, sum the areas:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Area from } l + \text{Area from } w
\]
\[
= 32 + 16\sqrt{5} + 8\sqrt{2}
\]
### 7. Convert Square Roots to Decimal Values
Now we can calculate numerical approximations:
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt{2} \approx 1.414 \)
Calculating:
\[
16\sqrt{5} \approx 16 \times 2.236 \approx 35.776 \, \text{cm}^2
\]
\[
8\sqrt{2} \approx 8 \times 1.414 \approx 11.312 \, \text{cm}^2
\]
### 8. Final Calculation
So the total surface area is:
\[
\text{Total Surface Area} \approx 32 + 35.776 + 11.312 \approx 79.088 \, \text{cm}^2
\]
### Rounding
Rounding to the nearest hundredth:
\[
\text{Total Surface Area} \approx 79.09 \, \text{cm}^2
\]
If you were seeking a different total or have other details leading to different conclusions, please let me know so we can align on the computations or clarify any misunderstandings! Thank you for your patience, and I'm here to assist.
To find the surface area \( A \) of a right rectangular pyramid, we need to do the following steps:
### 1. Surface Area Formula
The surface area of a right rectangular pyramid consists of the area of the base plus the areas of the triangular faces. The formula can be represented as:
\[
A = \text{Base Area} + \text{Lateral Area}
\]
### 2. Given Measurements
- Length of the base \( l = 8 \, \text{cm} \)
- Width of the base \( w = 4 \, \text{cm} \)
- Height of the pyramid \( h = 2 \, \text{cm} \)
### 3. Calculate the Base Area
The area of the rectangular base is:
\[
\text{Base Area} = l \times w = 8 \times 4 = 32 \, \text{cm}^2
\]
### 4. Calculate the Slant Heights
To find the slant heights of the triangular faces, we will need to determine the height of each triangular face using the Pythagorean theorem.
- **For triangular faces that have a base of length \( l \) (8 cm)**:
- Half of the base length is:
\[
\frac{l}{2} = \frac{8}{2} = 4 \, \text{cm}
\]
- The slant height \( h_1 \) can be calculated as:
\[
h_1 = \sqrt{\left(\frac{l}{2}\right)^2 + h^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \, \text{cm}
\]
- **For triangular faces that have a base of width \( w \) (4 cm)**:
- Half of the base width is:
\[
\frac{w}{2} = \frac{4}{2} = 2 \, \text{cm}
\]
- The slant height \( h_2 \) can be calculated as:
\[
h_2 = \sqrt{\left(\frac{w}{2}\right)^2 + h^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \, \text{cm}
\]
### 5. Calculate the Areas of the Triangular Faces
Now, calculate the areas of the triangular faces.
- **Area of the two triangles with base \( l \) (8 cm)**:
\[
\text{Area}_{l} = 2 \cdot \left(\frac{1}{2} \times l \times h_1\right) = 2 \cdot \left(\frac{1}{2} \times 8 \times 2\sqrt{5}\right)
\]
\[
= 2 \cdot (8\sqrt{5}) = 16\sqrt{5} \, \text{cm}^2
\]
- **Area of the two triangles with base \( w \) (4 cm)**:
\[
\text{Area}_{w} = 2 \cdot \left(\frac{1}{2} \times w \times h_2\right) = 2 \cdot \left(\frac{1}{2} \times 4 \times 2\sqrt{2}\right)
\]
\[
= 2 \cdot (4\sqrt{2}) = 8\sqrt{2} \, \text{cm}^2
\]
### 6. Calculate the Total Surface Area
Now, sum the areas:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Area from } l + \text{Area from } w
\]
\[
= 32 + 16\sqrt{5} + 8\sqrt{2}
\]
### 7. Convert Square Roots to Decimal Values
Now we can calculate numerical approximations:
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt{2} \approx 1.414 \)
Calculating:
\[
16\sqrt{5} \approx 16 \times 2.236 \approx 35.776 \, \text{cm}^2
\]
\[
8\sqrt{2} \approx 8 \times 1.414 \approx 11.312 \, \text{cm}^2
\]
### 8. Final Calculation
So the total surface area is:
\[
\text{Total Surface Area} \approx 32 + 35.776 + 11.312 \approx 79.088 \, \text{cm}^2
\]
### Rounding
Rounding to the nearest hundredth:
\[
\text{Total Surface Area} \approx 79.09 \, \text{cm}^2
\]
If you were seeking a different total or have other details leading to different conclusions, please let me know so we can align on the computations or clarify any misunderstandings! Thank you for your patience, and I'm here to assist.
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