Question

Find the surface area of a rectangular pyramid with these measurements: l
= 8 cm, w
= 4 cm, and h
= 2 cm

Answer 1

Find the surface area of a rectangular pyramid with these measurements: l
= 13 cm, w
= 11 cm, and h
= 8 cm

Answer 2

To find the surface area of a rectangular pyramid, we need to calculate the area of the base and the areas of the triangular faces.

Given measurements:
- Length (\( l \)) = 8 cm
- Width (\( w \)) = 4 cm
- Height (\( h \)) = 2 cm

### Step 1: Calculate the area of the base
The base of the pyramid is a rectangle, so the area \( A_{\text{base}} \) is calculated as:
\[
A_{\text{base}} = l \times w = 8 \, \text{cm} \times 4 \, \text{cm} = 32 \, \text{cm}^2
\]

### Step 2: Calculate the slant heights
To find the area of the triangular faces, we need the slant heights of the triangles. The slant height can be found using the Pythagorean theorem.

1. **Slant height along the length:**
The slant height for the triangular face along the length can be calculated as:
\[
l_s = \sqrt{\left(\frac{l}{2}\right)^2 + h^2} = \sqrt{\left(\frac{8 \, \text{cm}}{2}\right)^2 + (2 \, \text{cm})^2} = \sqrt{(4 \, \text{cm})^2 + (2 \, \text{cm})^2} = \sqrt{16 \, \text{cm}^2 + 4 \, \text{cm}^2} = \sqrt{20 \, \text{cm}^2}
\]
\[
l_s = \sqrt{20} \approx 4.47 \, \text{cm}
\]

2. **Slant height along the width:**
The slant height for the triangular face along the width is calculated similarly:
\[
w_s = \sqrt{\left(\frac{w}{2}\right)^2 + h^2} = \sqrt{\left(\frac{4 \, \text{cm}}{2}\right)^2 + (2 \, \text{cm})^2} = \sqrt{(2 \, \text{cm})^2 + (2 \, \text{cm})^2} = \sqrt{4 \, \text{cm}^2 + 4 \, \text{cm}^2} = \sqrt{8 \, \text{cm}^2}
\]
\[
w_s = \sqrt{8} \approx 2.83 \, \text{cm}
\]

### Step 3: Calculate the area of the triangular faces
The area of each triangular face can be calculated as:
- Two triangles with slant height along the length:
\[
A_{\text{triangular, l}} = \frac{1}{2} \times l \times l_s = \frac{1}{2} \times 8 \, \text{cm} \times 4.47 \, \text{cm} \approx 17.88 \, \text{cm}^2
\]
Total for two triangular faces:
\[
A_{\text{triangular, l, total}} = 2 \times A_{\text{triangular, l}} \approx 2 \times 17.88 \, \text{cm}^2 \approx 35.76 \, \text{cm}^2
\]

- Two triangles with slant height along the width:
\[
A_{\text{triangular, w}} = \frac{1}{2} \times w \times w_s = \frac{1}{2} \times 4 \, \text{cm} \times 2.83 \, \text{cm} \approx 5.66 \, \text{cm}^2
\]
Total for two triangular faces:
\[
A_{\text{triangular, w, total}} = 2 \times A_{\text{triangular, w}} \approx 2 \times 5.66 \, \text{cm}^2 \approx 11.32 \, \text{cm}^2
\]

### Step 4: Calculate the total surface area
The total surface area \( A_{\text{total}} \) of the pyramid is given by:
\[
A_{\text{total}} = A_{\text{base}} + A_{\text{triangular, l, total}} + A_{\text{triangular, w, total}}
\]
\[
A_{\text{total}} = 32 \, \text{cm}^2 + 35.76 \, \text{cm}^2 + 11.32 \, \text{cm}^2 \approx 79.08 \, \text{cm}^2
\]

Therefore, the surface area of the rectangular pyramid is approximately \( 79.08 \, \text{cm}^2 \).

Answer 3

To find the surface area of a rectangular pyramid, we will calculate the area of the base and the areas of the triangular faces.

Given measurements:
- Length (\( l \)) = 13 cm
- Width (\( w \)) = 11 cm
- Height (\( h \)) = 8 cm

### Step 1: Calculate the area of the base
The base of the pyramid is a rectangle, so the area \( A_{\text{base}} \) is calculated as:

\[
A_{\text{base}} = l \times w = 13 \, \text{cm} \times 11 \, \text{cm} = 143 \, \text{cm}^2
\]

### Step 2: Calculate the slant heights
To find the area of the triangular faces, we need the slant heights of the triangles. The slant height can be calculated using the Pythagorean theorem.

1. **Slant height along the length:**
The slant height \( l_s \) for the triangular face along the length is given by:

\[
l_s = \sqrt{\left(\frac{l}{2}\right)^2 + h^2} = \sqrt{\left(\frac{13 \, \text{cm}}{2}\right)^2 + (8 \, \text{cm})^2} = \sqrt{(6.5 \, \text{cm})^2 + (8 \, \text{cm})^2}
\]

\[
l_s = \sqrt{42.25 \, \text{cm}^2 + 64 \, \text{cm}^2} = \sqrt{106.25 \, \text{cm}^2} \approx 10.31 \, \text{cm}
\]

2. **Slant height along the width:**
The slant height \( w_s \) for the triangular face along the width is calculated as:

\[
w_s = \sqrt{\left(\frac{w}{2}\right)^2 + h^2} = \sqrt{\left(\frac{11 \, \text{cm}}{2}\right)^2 + (8 \, \text{cm})^2} = \sqrt{(5.5 \, \text{cm})^2 + (8 \, \text{cm})^2}
\]

\[
w_s = \sqrt{30.25 \, \text{cm}^2 + 64 \, \text{cm}^2} = \sqrt{94.25 \, \text{cm}^2} \approx 9.71 \, \text{cm}
\]

### Step 3: Calculate the area of the triangular faces
The area of each triangular face can be calculated using the formula for the area of a triangle (\( \frac{1}{2} \times \text{base} \times \text{height} \)).

1. **Area of the two triangular faces along the length:**

\[
A_{\text{triangular, l}} = \frac{1}{2} \times l \times l_s = \frac{1}{2} \times 13 \, \text{cm} \times 10.31 \, \text{cm} \approx 67.08 \, \text{cm}^2
\]

Total for two triangular faces (along the length):

\[
A_{\text{triangular, l, total}} = 2 \times A_{\text{triangular, l}} \approx 2 \times 67.08 \, \text{cm}^2 \approx 134.16 \, \text{cm}^2
\]

2. **Area of the two triangular faces along the width:**

\[
A_{\text{triangular, w}} = \frac{1}{2} \times w \times w_s = \frac{1}{2} \times 11 \, \text{cm} \times 9.71 \, \text{cm} \approx 53.36 \, \text{cm}^2
\]

Total for two triangular faces (along the width):

\[
A_{\text{triangular, w, total}} = 2 \times A_{\text{triangular, w}} \approx 2 \times 53.36 \, \text{cm}^2 \approx 106.72 \, \text{cm}^2
\]

### Step 4: Calculate the total surface area
The total surface area \( A_{\text{total}} \) of the pyramid is given by:

\[
A_{\text{total}} = A_{\text{base}} + A_{\text{triangular, l, total}} + A_{\text{triangular, w, total}}
\]

\[
A_{\text{total}} = 143 \, \text{cm}^2 + 134.16 \, \text{cm}^2 + 106.72 \, \text{cm}^2 \approx 383.88 \, \text{cm}^2
\]

Therefore, the surface area of the rectangular pyramid is approximately \( 383.88 \, \text{cm}^2 \).