Question

Find the TOTAL surface area of this triangular prism in ``cm2
Here are the measurements 5cm 5cm 6cm 4cm 12cm

Answer 1

To find the total surface area of the triangular prism, we first need to understand the dimensions given. It looks like you have provided the side lengths of the triangular base (5 cm, 5 cm, and 6 cm) and the height (or length) of the prism, which is 12 cm.

### Step 1: Calculate the area of the triangular base

To calculate the area of the triangular base, we can use Heron's formula. First, we find the semi-perimeter (s) of the triangle.

The sides of the triangle are:
- \(a = 5 \, \text{cm}\)
- \(b = 5 \, \text{cm}\)
- \(c = 6 \, \text{cm}\)

#### Calculate the semi-perimeter:
\[
s = \frac{a + b + c}{2} = \frac{5 + 5 + 6}{2} = 8 \, \text{cm}
\]

#### Apply Heron's formula to find the area (A) of the triangle:
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
\[
A = \sqrt{8(8-5)(8-5)(8-6)} = \sqrt{8 \times 3 \times 3 \times 2}
\]
\[
= \sqrt{144} = 12 \, \text{cm}^2
\]

### Step 2: Calculate the lateral surface area

The lateral surface area of the prism is calculated by finding the area of the three rectangular faces. The height of the prism is 12 cm, and we multiply the perimeter of the triangular base by the height.

#### Calculate the perimeter of the triangular base:
\[
\text{Perimeter} = a + b + c = 5 + 5 + 6 = 16 \, \text{cm}
\]

#### Lateral surface area (LSA):
\[
\text{LSA} = \text{Perimeter} \times \text{Height} = 16 \times 12 = 192 \, \text{cm}^2
\]

### Step 3: Total surface area

The total surface area of the prism includes the lateral area plus the areas of the two triangular bases.

#### Total surface area (TSA):
\[
\text{TSA} = \text{LSA} + 2 \times \text{Area of the base}
\]
\[
\text{TSA} = 192 + 2 \times 12 = 192 + 24 = 216 \, \text{cm}^2
\]

### Final Answer:
The total surface area of the triangular prism is \( \boxed{216 \, \text{cm}^2} \).