Question
Surface Area of Pyramids Practice
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Question
Find the surface area of a rectangular pyramid with these measurements: l
= 8 cm, w
= 4 cm, and h
= 2 cm. Express your answer as a decimal rounded to the nearest hundredth.(1 point)
cm2
Complete this assessment to review what you've learned. It will not count toward your grade.
1 of 51 of 5 Items
Question
Find the surface area of a rectangular pyramid with these measurements: l
= 8 cm, w
= 4 cm, and h
= 2 cm. Express your answer as a decimal rounded to the nearest hundredth.(1 point)
cm2
Answers
GPT-4o mini
To find the surface area of a rectangular pyramid, you can use the formula:
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area}
\]
1. **Calculate the Base Area:**
The base of the pyramid is a rectangle, and its area can be calculated using:
\[
\text{Base Area} = l \times w = 8 \, \text{cm} \times 4 \, \text{cm} = 32 \, \text{cm}^2
\]
2. **Calculate the Slant Height:**
To find the lateral area, we first need the slant height (\(s\)). The slant height can be calculated using the Pythagorean theorem. The height (\(h\)) is given as 2 cm, and since the base is 8 cm long, we need half of the base length:
\[
\text{Half base} = \frac{l}{2} = \frac{8}{2} = 4 \, \text{cm}
\]
Now we can find the slant height (\(s\)):
\[
s = \sqrt{h^2 + \left(\frac{l}{2}\right)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \approx 4.47 \, \text{cm}
\]
3. **Calculate the Lateral Area:**
The lateral area consists of the areas of the four triangular faces. The area of each triangular face can be calculated using the formula:
\[
\text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the two triangles with base 8 cm:
\[
\text{Area} = \frac{1}{2} \times 8 \times 4.47 \approx 17.88 \, \text{cm}^2
\]
For the two triangles with base 4 cm:
\[
\text{Area} = \frac{1}{2} \times 4 \times 4.47 \approx 8.94 \, \text{cm}^2
\]
So the total lateral area is:
\[
\text{Total Lateral Area} = 2 \times 17.88 + 2 \times 8.94 = 35.76 + 17.88 = 53.64 \, \text{cm}^2
\]
4. **Calculate Total Surface Area:**
Now add the base area to the lateral area:
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area} = 32 + 53.64 = 85.64 \, \text{cm}^2
\]
Thus, the surface area of the rectangular pyramid is:
\[
\boxed{85.64 \, \text{cm}^2}
\]
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area}
\]
1. **Calculate the Base Area:**
The base of the pyramid is a rectangle, and its area can be calculated using:
\[
\text{Base Area} = l \times w = 8 \, \text{cm} \times 4 \, \text{cm} = 32 \, \text{cm}^2
\]
2. **Calculate the Slant Height:**
To find the lateral area, we first need the slant height (\(s\)). The slant height can be calculated using the Pythagorean theorem. The height (\(h\)) is given as 2 cm, and since the base is 8 cm long, we need half of the base length:
\[
\text{Half base} = \frac{l}{2} = \frac{8}{2} = 4 \, \text{cm}
\]
Now we can find the slant height (\(s\)):
\[
s = \sqrt{h^2 + \left(\frac{l}{2}\right)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \approx 4.47 \, \text{cm}
\]
3. **Calculate the Lateral Area:**
The lateral area consists of the areas of the four triangular faces. The area of each triangular face can be calculated using the formula:
\[
\text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the two triangles with base 8 cm:
\[
\text{Area} = \frac{1}{2} \times 8 \times 4.47 \approx 17.88 \, \text{cm}^2
\]
For the two triangles with base 4 cm:
\[
\text{Area} = \frac{1}{2} \times 4 \times 4.47 \approx 8.94 \, \text{cm}^2
\]
So the total lateral area is:
\[
\text{Total Lateral Area} = 2 \times 17.88 + 2 \times 8.94 = 35.76 + 17.88 = 53.64 \, \text{cm}^2
\]
4. **Calculate Total Surface Area:**
Now add the base area to the lateral area:
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area} = 32 + 53.64 = 85.64 \, \text{cm}^2
\]
Thus, the surface area of the rectangular pyramid is:
\[
\boxed{85.64 \, \text{cm}^2}
\]
That is no correct, you can do this
GPT-4o mini
Let’s carefully recalculate the surface area of the rectangular pyramid with the given measurements: length \( l = 8 \) cm, width \( w = 4 \) cm, and height \( h = 2 \) cm.
### Step 1: Calculate the Base Area
The base area \( A_b \) of the rectangular pyramid is calculated as follows:
\[
A_b = l \times w = 8 \, \text{cm} \times 4 \, \text{cm} = 32 \, \text{cm}^2
\]
### Step 2: Calculate the Slant Height
Next, we need the slant height \( s \) to calculate the lateral surface area. The slant height can be found using the Pythagorean theorem. The height \( h \) is 2 cm, and we use half of the length for one of the triangles:
To find \( s \):
- Half of the base length \( l \):
\[
\text{Half base} = \frac{l}{2} = \frac{8}{2} = 4 \, \text{cm}
\]
Use the Pythagorean theorem:
\[
s = \sqrt{h^2 + \left(\frac{l}{2}\right)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \approx 4.47 \, \text{cm}
\]
### Step 3: Calculate the Lateral Area
Now, calculate the lateral area of the pyramid which consists of four triangular faces. The lateral area \( A_L \) can be calculated as follows:
1. For the triangles with base \( l = 8 \, \text{cm} \):
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 8 \times 4.47 \approx 17.88 \, \text{cm}^2
\]
2. For the triangles with base \( w = 4 \, \text{cm} \):
\[
\text{Area of one triangle} = \frac{1}{2} \times 4 \times 4.47 \approx 8.94 \, \text{cm}^2
\]
There are two triangles of each type, so the total lateral area becomes:
\[
A_L = 2 \times 17.88 + 2 \times 8.94 = 35.76 + 17.88 \approx 53.64 \, \text{cm}^2
\]
### Step 4: Calculate the Total Surface Area
Now we add the base area and the lateral area:
\[
\text{Total Surface Area} = A_b + A_L = 32 \, \text{cm}^2 + 53.64 \, \text{cm}^2 = 85.64 \, \text{cm}^2
\]
### Final Answer
\[
\boxed{85.64 \, \text{cm}^2}
\]
If you meant to calculate a different surface area or if there are specific parameters I should be aware of, please let me know!
### Step 1: Calculate the Base Area
The base area \( A_b \) of the rectangular pyramid is calculated as follows:
\[
A_b = l \times w = 8 \, \text{cm} \times 4 \, \text{cm} = 32 \, \text{cm}^2
\]
### Step 2: Calculate the Slant Height
Next, we need the slant height \( s \) to calculate the lateral surface area. The slant height can be found using the Pythagorean theorem. The height \( h \) is 2 cm, and we use half of the length for one of the triangles:
To find \( s \):
- Half of the base length \( l \):
\[
\text{Half base} = \frac{l}{2} = \frac{8}{2} = 4 \, \text{cm}
\]
Use the Pythagorean theorem:
\[
s = \sqrt{h^2 + \left(\frac{l}{2}\right)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \approx 4.47 \, \text{cm}
\]
### Step 3: Calculate the Lateral Area
Now, calculate the lateral area of the pyramid which consists of four triangular faces. The lateral area \( A_L \) can be calculated as follows:
1. For the triangles with base \( l = 8 \, \text{cm} \):
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 8 \times 4.47 \approx 17.88 \, \text{cm}^2
\]
2. For the triangles with base \( w = 4 \, \text{cm} \):
\[
\text{Area of one triangle} = \frac{1}{2} \times 4 \times 4.47 \approx 8.94 \, \text{cm}^2
\]
There are two triangles of each type, so the total lateral area becomes:
\[
A_L = 2 \times 17.88 + 2 \times 8.94 = 35.76 + 17.88 \approx 53.64 \, \text{cm}^2
\]
### Step 4: Calculate the Total Surface Area
Now we add the base area and the lateral area:
\[
\text{Total Surface Area} = A_b + A_L = 32 \, \text{cm}^2 + 53.64 \, \text{cm}^2 = 85.64 \, \text{cm}^2
\]
### Final Answer
\[
\boxed{85.64 \, \text{cm}^2}
\]
If you meant to calculate a different surface area or if there are specific parameters I should be aware of, please let me know!