Surface Area using Nets Worksheets - Free Printable

Let's solve each problem step by step using the nets of the solid shapes. We'll calculate the surface area by finding the area of each individual face in the net and then adding them together.

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Problem 1: Triangular Prism

#### Given Net:
- The solid is a triangular prism.
- The net shows:
- Two triangular faces (J and L)
- Three rectangular faces (K, M, H)

#### Dimensions:
From the diagram:
- Triangle base = 6 ft
- Height of triangle = 4 ft
- Side lengths of triangle: 5 ft, 5 ft, 6 ft (isosceles triangle)
- Rectangular sides:
- K: 6 ft × 7 ft
- M: 5 ft × 7 ft
- H: 5 ft × 7 ft

> Note: The height of the prism (depth) is 7 ft.

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#### Step-by-step Area Calculations:

Area of J (triangle):
$$
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 4 = 12 \text{ ft}^2
$$

Area of L (other triangle):
Same as J → $12 \text{ ft}^2$

Area of K (rectangle):
$$
6 \times 7 = 42 \text{ ft}^2
$$

Area of M (rectangle):
$$
5 \times 7 = 35 \text{ ft}^2
$$

Area of H (rectangle):
$$
5 \times 7 = 35 \text{ ft}^2
$$

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#### Total Surface Area:
$$
12 + 12 + 42 + 35 + 35 = 136 \text{ ft}^2
$$

✔ Answer for Problem 1:
- Area of J = 12 ft²
- Area of K = 42 ft²
- Area of L = 12 ft²
- Area of M = 35 ft²
- Area of N = 35 ft² *(Note: Probably typo – should be H instead of N)*
- Surface Area = 136 ft²

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Problem 2: Square Pyramid

#### Given Net:
- One square base (W)
- Four triangular faces (U, V, X, Y)

#### Dimensions:
- Base side = 10 yd
- Slant height of triangle = 12 yd
- Height of pyramid = 7.2 yd (not needed — we use slant height for lateral faces)

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#### Step-by-step Area Calculations:

Area of U (triangle):
$$
\frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 10 \times 12 = 60 \text{ yd}^2
$$

Similarly, all four triangular faces are congruent:

Area of V = 60 yd²
Area of X = 60 yd²
Area of Y = 60 yd²

Area of W (square base):
$$
10 \times 10 = 100 \text{ yd}^2
$$

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#### Total Surface Area:
$$
60 + 60 + 60 + 60 + 100 = 400 \text{ yd}^2
$$

✔ Answer for Problem 2:
- Area of U = 60 yd²
- Area of V = 60 yd²
- Area of W = 100 yd²
- Area of X = 60 yd²
- Area of Y = 60 yd²
- Surface Area = 400 yd²

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Problem 3: Rectangular Prism (Cuboid)

#### Given Net:
- Six rectangles: A, B, C, D, E, F
- Dimensions from diagram:
- Length = 10 m
- Width = 4 m
- Height = 12 m

The net shows:
- A and F: top and bottom → 10 m × 4 m
- B and E: front and back → 10 m × 12 m
- C and D: left and right → 4 m × 12 m

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#### Step-by-step Area Calculations:

Area of A (top):
$$
10 \times 4 = 40 \text{ m}^2
$$

Area of B (front):
$$
10 \times 12 = 120 \text{ m}^2
$$

Area of C (left side):
$$
4 \times 12 = 48 \text{ m}^2
$$

Area of D (right side):
$$
4 \times 12 = 48 \text{ m}^2
$$

Area of E (back):
$$
10 \times 12 = 120 \text{ m}^2
$$

Area of F (bottom):
$$
10 \times 4 = 40 \text{ m}^2
$$

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#### Total Surface Area:
$$
40 + 120 + 48 + 48 + 120 + 40 = 416 \text{ m}^2
$$

✔ Answer for Problem 3:
- Area of A = 40 m²
- Area of B = 120 m²
- Area of C = 48 m²
- Area of D = 48 m²
- Area of E = 120 m²
- Area of F = 40 m²
- Surface Area = 416 m²

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✔ Final Answers Summary:

#### Problem 1: Triangular Prism
- Area of J = 12 ft²
- Area of K = 42 ft²
- Area of L = 12 ft²
- Area of M = 35 ft²
- Area of N = 35 ft² *(assuming N is H)*
- Surface Area = 136 ft²

#### Problem 2: Square Pyramid
- Area of U = 60 yd²
- Area of V = 60 yd²
- Area of W = 100 yd²
- Area of X = 60 yd²
- Area of Y = 60 yd²
- Surface Area = 400 yd²

#### Problem 3: Rectangular Prism
- Area of A = 40 m²
- Area of B = 120 m²
- Area of C = 48 m²
- Area of D = 48 m²
- Area of E = 120 m²
- Area of F = 40 m²
- Surface Area = 416 m²

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