Question
Find the TOTAL surface area of this triangular prism in cm^2.
5 cm, 6 cm, 4 cm, 12 cm, 5 cm.
5 cm, 6 cm, 4 cm, 12 cm, 5 cm.
Answers
Answered by
GPT-4o mini
To find the total surface area of the triangular prism, we need to calculate the areas of the two triangular bases and the three rectangular faces.
Given the dimensions:
- The dimensions of the triangular base are 5 cm, 6 cm, and 4 cm.
- The height (length) of the prism is 12 cm.
### Step 1: Calculate the area of the triangular base.
1. **Find the semi-perimeter \(s\)**:
\[
s = \frac{a + b + c}{2} = \frac{5 + 6 + 4}{2} = \frac{15}{2} = 7.5 \text{ cm}
\]
2. **Use Heron's formula to find the area \(A\)**:
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
Where \( a = 5 \) cm, \( b = 6 \) cm, and \( c = 4 \) cm.
Calculating each part:
\[
s - a = 7.5 - 5 = 2.5
\]
\[
s - b = 7.5 - 6 = 1.5
\]
\[
s - c = 7.5 - 4 = 3.5
\]
Substituting values into Heron's formula:
\[
A = \sqrt{7.5 \times 2.5 \times 1.5 \times 3.5}
\]
Calculating step-by-step:
\[
7.5 \times 2.5 = 18.75
\]
\[
1.5 \times 3.5 = 5.25
\]
\[
18.75 \times 5.25 = 98.4375
\]
Thus,
\[
A = \sqrt{98.4375} \approx 9.92 \text{ cm}^2
\]
### Step 2: Calculate the total area of two triangular bases.
Since there are two identical triangular bases:
\[
\text{Area of two bases} = 2 \times A \approx 2 \times 9.92 \approx 19.84 \text{ cm}^2
\]
### Step 3: Calculate the areas of the rectangular faces.
The dimensions for the three rectangular faces are:
1. Rectangular face connected to side of 5 cm: Area \( = 5 \times 12 = 60 \, \text{cm}^2 \)
2. Rectangular face connected to side of 6 cm: Area \( = 6 \times 12 = 72 \, \text{cm}^2 \)
3. Rectangular face connected to side of 4 cm: Area \( = 4 \times 12 = 48 \, \text{cm}^2 \)
### Total area of rectangular faces:
\[
\text{Total area of rectangular faces} = 60 + 72 + 48 = 180 \text{ cm}^2
\]
### Step 4: Calculate the total surface area of the prism.
Finally, the total surface area \(S\) of the prism is:
\[
S = \text{Area of two bases} + \text{Area of rectangular faces}
\]
\[
S = 19.84 + 180 = 199.84 \text{ cm}^2
\]
Thus, the total surface area of the triangular prism is approximately:
\[
\boxed{199.84 \text{ cm}^2}
\]
Given the dimensions:
- The dimensions of the triangular base are 5 cm, 6 cm, and 4 cm.
- The height (length) of the prism is 12 cm.
### Step 1: Calculate the area of the triangular base.
1. **Find the semi-perimeter \(s\)**:
\[
s = \frac{a + b + c}{2} = \frac{5 + 6 + 4}{2} = \frac{15}{2} = 7.5 \text{ cm}
\]
2. **Use Heron's formula to find the area \(A\)**:
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
Where \( a = 5 \) cm, \( b = 6 \) cm, and \( c = 4 \) cm.
Calculating each part:
\[
s - a = 7.5 - 5 = 2.5
\]
\[
s - b = 7.5 - 6 = 1.5
\]
\[
s - c = 7.5 - 4 = 3.5
\]
Substituting values into Heron's formula:
\[
A = \sqrt{7.5 \times 2.5 \times 1.5 \times 3.5}
\]
Calculating step-by-step:
\[
7.5 \times 2.5 = 18.75
\]
\[
1.5 \times 3.5 = 5.25
\]
\[
18.75 \times 5.25 = 98.4375
\]
Thus,
\[
A = \sqrt{98.4375} \approx 9.92 \text{ cm}^2
\]
### Step 2: Calculate the total area of two triangular bases.
Since there are two identical triangular bases:
\[
\text{Area of two bases} = 2 \times A \approx 2 \times 9.92 \approx 19.84 \text{ cm}^2
\]
### Step 3: Calculate the areas of the rectangular faces.
The dimensions for the three rectangular faces are:
1. Rectangular face connected to side of 5 cm: Area \( = 5 \times 12 = 60 \, \text{cm}^2 \)
2. Rectangular face connected to side of 6 cm: Area \( = 6 \times 12 = 72 \, \text{cm}^2 \)
3. Rectangular face connected to side of 4 cm: Area \( = 4 \times 12 = 48 \, \text{cm}^2 \)
### Total area of rectangular faces:
\[
\text{Total area of rectangular faces} = 60 + 72 + 48 = 180 \text{ cm}^2
\]
### Step 4: Calculate the total surface area of the prism.
Finally, the total surface area \(S\) of the prism is:
\[
S = \text{Area of two bases} + \text{Area of rectangular faces}
\]
\[
S = 19.84 + 180 = 199.84 \text{ cm}^2
\]
Thus, the total surface area of the triangular prism is approximately:
\[
\boxed{199.84 \text{ cm}^2}
\]