Calculate the surface area of triangular prisms using provided dimensions and net diagrams.

We are given three triangular prisms and asked to calculate their total surface areas.

A triangular prism has:
- 2 triangular bases (identical)
- 3 rectangular lateral faces

The total surface area (SA) = Area of 2 triangles + Area of 3 rectangles

---

## 🔹 Problem 1)

Given:

- Two triangular bases: right triangles with legs 5m and 8m
- Three rectangular faces:
- One rectangle: 7m × 9.4m
- One rectangle: 7m × 8m
- One rectangle: 7m × 5m

*(Note: The side lengths of the triangle are 5m, 8m, and hypotenuse = √(5²+8²) = √(25+64)=√89 ≈ 9.43m — which matches the 9.4m shown. So the triangle is right-angled.)*

Step 1: Area of one triangle

Area = (1/2) × base × height = (1/2) × 5 × 8 = 20 m²

→ Two triangles: 2 × 20 = 40 m²

Step 2: Areas of the three rectangles

- Rectangle 1: 7m × 9.4m = 65.8 m²
- Rectangle 2: 7m × 8m = 56 m²
- Rectangle 3: 7m × 5m = 35 m²

→ Total rectangles = 65.8 + 56 + 35 = 156.8 m²

✔ Total Surface Area = 40 + 156.8 = 196.8 m²

---

## 🔹 Problem 2)

Given:

- Triangular base: sides 6m, 5m, and 12m? Wait — that can’t be a triangle! 6 + 5 = 11 < 12 → violates triangle inequality.

Wait — let’s re-examine.

Actually, looking at the diagram:

It shows a triangular prism with:

- Triangle sides: 6m, 5m, and hypotenuse 13.4m? But 6² + 5² = 36 + 25 = 61 → √61 ≈ 7.81, not 13.4.

But wait — the vertical side is labeled 12m, and base is 6m, and slant side is 13.4m.

Check: 6² + 12² = 36 + 144 = 180 → √180 ≈ 13.416 → so yes, it’s a right triangle with legs 6m and 12m, hypotenuse ≈13.4m.

So triangle is right-angled with legs 6m and 12m.

Step 1: Area of one triangle

= (1/2) × 6 × 12 = 36 m²

→ Two triangles: 2 × 36 = 72 m²

Step 2: Rectangles

The three rectangles have heights equal to the prism length — which is 5m (the depth shown).

Their widths are the sides of the triangle: 6m, 12m, and 13.4m.

So:

- Rectangle 1: 5m × 6m = 30 m²
- Rectangle 2: 5m × 12m = 60 m²
- Rectangle 3: 5m × 13.4m = 67 m²

→ Total rectangles = 30 + 60 + 67 = 157 m²

✔ Total Surface Area = 72 + 157 = 229 m²

---

## 🔹 Problem 3)

Given:

- Triangular base: sides 12m, 10m, and 10m → this is an isosceles triangle.
- Height of prism (depth) = 15.6m? Wait — no, 15.6m is the slant side of the triangle? Let’s check.

Actually, looking at the diagram:

- Base of triangle: 12m
- Two equal sides: 10m each
- The height of the prism (the length perpendicular to the triangle) is 15.6m

Wait — the label “15.6m” is on the rectangular face, meaning it's the length of the prism, i.e., the distance between the two triangular bases.

So:

- Triangular base: isosceles triangle with base 12m, equal sides 10m.
- Prism length (height) = 15.6m

We need to find the area of the triangle first.

Step 1: Area of the triangle

Since it’s isosceles with base 12m and sides 10m, we can find the height using Pythagoras.

Split the base into two 6m segments.

Height h = √(10² - 6²) = √(100 - 36) = √64 = 8m

→ Area of one triangle = (1/2) × 12 × 8 = 48 m²

→ Two triangles: 2 × 48 = 96 m²

Step 2: Rectangles

Three rectangles, each with height = 15.6m, and widths = sides of triangle: 12m, 10m, 10m.

So:

- Rectangle 1: 15.6 × 12 = 187.2 m²
- Rectangle 2: 15.6 × 10 = 156 m²
- Rectangle 3: 15.6 × 10 = 156 m²

→ Total rectangles = 187.2 + 156 + 156 = 499.2 m²

✔ Total Surface Area = 96 + 499.2 = 595.2 m²

---

## 📦 Final Answers:

1) 196.8 m²
2) 229 m²
3) 595.2 m²

✔ All solved with clear steps and reasoning.