Question
ind the surface area of a rectangular pyramid with these measurements: l
= 13 cm, w
= 11 cm, and h
= 8 cm. Express your answer as a decimal rounded to the nearest hundredth.(1 point)
= 13 cm, w
= 11 cm, and h
= 8 cm. Express your answer as a decimal rounded to the nearest hundredth.(1 point)
Answers
GPT-4o mini
To find the surface area of a rectangular pyramid, we need to calculate the area of the base and the area of the four triangular sides (the lateral surface area).
### Step 1: Area of the Base
The base of the pyramid is a rectangle, so we can find the area of the base using the formula:
\[
\text{Area of the base} = l \times w
\]
Given:
- \( l = 13 \) cm
- \( w = 11 \) cm
Calculating the area of the base:
\[
\text{Area of the base} = 13 \times 11 = 143 \, \text{cm}^2
\]
### Step 2: Lateral Surface Area
To find the lateral surface area, we need to find the area of the four triangular faces. Each triangular face has a base equal to one of the sides of the rectangular base and a height that can be found using the slant height of the pyramid.
First, we need to find the slant height. The height from the apex of the pyramid down to the center of the base can be used with the half-lengths of the base to determine the slant height.
- The half-length (\( a \)) and half-width (\( b \)) are:
- \( a = \frac{l}{2} = \frac{13}{2} = 6.5 \) cm
- \( b = \frac{w}{2} = \frac{11}{2} = 5.5 \) cm
The slant height \( s \) can be found using the Pythagorean theorem:
\[
s = \sqrt{h^2 + a^2} = \sqrt{8^2 + 6.5^2}
\]
Calculating each term:
\[
h^2 = 64
\]
\[
a^2 = 6.5^2 = 42.25
\]
\[
s = \sqrt{64 + 42.25} = \sqrt{106.25} \approx 10.31 \, \text{cm}
\]
Now, we can calculate the area of each triangular face:
1. **Triangles with base 13 cm:**
\[
\text{Area}_{\text{triangle 1}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 13 \times 10.31 \approx 67.01 \, \text{cm}^2
\]
2. **Triangles with base 11 cm:**
\[
\text{Area}_{\text{triangle 2}} = \frac{1}{2} \times 11 \times 10.31 \approx 56.71 \, \text{cm}^2
\]
Since there are two triangles of each type:
\[
\text{Total lateral area} = 2 \times \text{Area}_{\text{triangle 1}} + 2 \times \text{Area}_{\text{triangle 2}}
\]
Calculating total lateral area:
\[
\text{Total lateral area} = 2(67.01) + 2(56.71) = 134.02 + 113.42 = 247.44 \, \text{cm}^2
\]
### Step 3: Total Surface Area
Finally, the total surface area is the sum of the base area and total lateral area:
\[
\text{Surface Area} = \text{Area of the base} + \text{Total lateral area}
\]
Calculating the total surface area:
\[
\text{Surface Area} = 143 + 247.44 = 390.44 \, \text{cm}^2
\]
Thus, the total surface area of the rectangular pyramid is:
\[
\boxed{390.44} \, \text{cm}^2
\]
### Step 1: Area of the Base
The base of the pyramid is a rectangle, so we can find the area of the base using the formula:
\[
\text{Area of the base} = l \times w
\]
Given:
- \( l = 13 \) cm
- \( w = 11 \) cm
Calculating the area of the base:
\[
\text{Area of the base} = 13 \times 11 = 143 \, \text{cm}^2
\]
### Step 2: Lateral Surface Area
To find the lateral surface area, we need to find the area of the four triangular faces. Each triangular face has a base equal to one of the sides of the rectangular base and a height that can be found using the slant height of the pyramid.
First, we need to find the slant height. The height from the apex of the pyramid down to the center of the base can be used with the half-lengths of the base to determine the slant height.
- The half-length (\( a \)) and half-width (\( b \)) are:
- \( a = \frac{l}{2} = \frac{13}{2} = 6.5 \) cm
- \( b = \frac{w}{2} = \frac{11}{2} = 5.5 \) cm
The slant height \( s \) can be found using the Pythagorean theorem:
\[
s = \sqrt{h^2 + a^2} = \sqrt{8^2 + 6.5^2}
\]
Calculating each term:
\[
h^2 = 64
\]
\[
a^2 = 6.5^2 = 42.25
\]
\[
s = \sqrt{64 + 42.25} = \sqrt{106.25} \approx 10.31 \, \text{cm}
\]
Now, we can calculate the area of each triangular face:
1. **Triangles with base 13 cm:**
\[
\text{Area}_{\text{triangle 1}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 13 \times 10.31 \approx 67.01 \, \text{cm}^2
\]
2. **Triangles with base 11 cm:**
\[
\text{Area}_{\text{triangle 2}} = \frac{1}{2} \times 11 \times 10.31 \approx 56.71 \, \text{cm}^2
\]
Since there are two triangles of each type:
\[
\text{Total lateral area} = 2 \times \text{Area}_{\text{triangle 1}} + 2 \times \text{Area}_{\text{triangle 2}}
\]
Calculating total lateral area:
\[
\text{Total lateral area} = 2(67.01) + 2(56.71) = 134.02 + 113.42 = 247.44 \, \text{cm}^2
\]
### Step 3: Total Surface Area
Finally, the total surface area is the sum of the base area and total lateral area:
\[
\text{Surface Area} = \text{Area of the base} + \text{Total lateral area}
\]
Calculating the total surface area:
\[
\text{Surface Area} = 143 + 247.44 = 390.44 \, \text{cm}^2
\]
Thus, the total surface area of the rectangular pyramid is:
\[
\boxed{390.44} \, \text{cm}^2
\]