Finding Surface Area Using Nets #2 - Practice calculating surface area of 3D shapes using their nets.

Let's solve each problem step by step using the nets provided. The goal is to find the surface area of each 3D shape by calculating the total area of all the faces shown in the net.

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Problem 1: Rectangular Prism

Net Description:
- A rectangular prism with dimensions:
- Length = 3 yd
- Width = 3 yd
- Height = 3 yd

This is a cube since all sides are equal (3 yd).

#### Step-by-step:

A cube has 6 faces, each face being a square of side 3 yd.

Area of one face = $ 3 \times 3 = 9 $ sq yd
Total surface area = $ 6 \times 9 = 54 $ sq yd

✔ Surface Area = 54 square yards

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Problem 2: Rectangular Prism (not a cube)

Dimensions from the net:
- Length = 7 cm
- Width = 4 cm
- Height = 2 cm

The net shows:
- Two 7×4 faces (top and bottom)
- Two 7×2 faces (front and back)
- Two 4×2 faces (sides)

Let’s compute each:

1. Top/Bottom: $ 7 \times 4 = 28 $ cm² → two of them → $ 2 \times 28 = 56 $
2. Front/Back: $ 7 \times 2 = 14 $ cm² → two → $ 2 \times 14 = 28 $
3. Sides: $ 4 \times 2 = 8 $ cm² → two → $ 2 \times 8 = 16 $

Total surface area = $ 56 + 28 + 16 = 100 $ cm²

✔ Surface Area = 100 square centimeters

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Problem 3: Triangular Prism

Net Description:
- Two triangular bases
- Three rectangular lateral faces

From the diagram:
- Triangle base: base = 6 m, height = 10 m
- Rectangles: lengths are 13 m, 10 m, and 6 m (the three sides of the triangle), and height = 8 m (depth of prism)

#### Step 1: Area of triangular bases

Area of one triangle = $ \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 10 = 30 $ m²
Two triangles → $ 2 \times 30 = 60 $ m²

#### Step 2: Areas of rectangular faces

Each rectangle has height 8 m and width equal to a side of the triangle.

- Rectangle 1: $ 13 \times 8 = 104 $ m²
- Rectangle 2: $ 10 \times 8 = 80 $ m²
- Rectangle 3: $ 6 \times 8 = 48 $ m²

Sum of rectangles = $ 104 + 80 + 48 = 232 $ m²

#### Total Surface Area:

$ 60 + 232 = 292 $ m²

✔ Surface Area = 292 square meters

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Problem 4: Square Pyramid

Net Description:
- One square base
- Four triangular faces

From the diagram:
- Base: square with side = 10 in
- Each triangular face has base = 10 in, height = 9 in (slant height)

#### Step 1: Area of the square base

$ 10 \times 10 = 100 $ in²

#### Step 2: Area of one triangular face

$ \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 9 = 45 $ in²
Four such faces → $ 4 \times 45 = 180 $ in²

#### Total Surface Area:

$ 100 + 180 = 280 $ in²

✔ Surface Area = 280 square inches

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✔ Final Answers:

1. 54 square yards
2. 100 square centimeters
3. 292 square meters
4. 280 square inches

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