Surface Area of Prisms worksheet featuring problems to calculate the area of nets and surface area of 3D prisms.

Here are the step-by-step solutions for the problems on the worksheet.

Section A: Work out the area of the nets

To find the total area, we calculate the area of each shape in the net and add them together.
*Area of a rectangle = length × width*
*Area of a triangle = ½ × base × height*

1) Rectangular Prism Net
* Central Rectangle: $10 \text{ cm} \times 2 \text{ cm} = 20 \text{ cm}^2$
* Top & Bottom Flaps: There are two rectangles attached to the top and bottom. Their width is the same as the central one ($10 \text{ cm}$) and their height is given by the side flap ($2 \text{ cm}$).
Area = $2 \times (10 \text{ cm} \times 2 \text{ cm}) = 40 \text{ cm}^2$
* Side Flaps: There are two rectangles on the left and right. Their height is $2 \text{ cm}$ and their width is also $2 \text{ cm}$ (since the prism has a square cross-section of $2\times2$).
Area = $2 \times (2 \text{ cm} \times 2 \text{ cm}) = 8 \text{ cm}^2$
* Total Area: $20 + 40 + 8 = \mathbf{68 \text{ cm}^2}$

2) Triangular Prism Net
* Three Rectangles: The long sides of the prism form three rectangles. They all have a length of $12 \text{ cm}$. Their widths correspond to the sides of the triangle ($5 \text{ cm}$, $6 \text{ cm}$, and $5 \text{ cm}$).
Total Rectangle Area = $(5 + 6 + 5) \times 12 = 16 \times 12 = 192 \text{ cm}^2$
* Two Triangles: The bases are triangles with a base of $6 \text{ cm}$ and a height of $4 \text{ cm}$.
Area of one triangle = $0.5 \times 6 \times 4 = 12 \text{ cm}^2$
Area of two triangles = $2 \times 12 = 24 \text{ cm}^2$
* Total Area: $192 + 24 = \mathbf{216 \text{ cm}^2}$

3) Pentagonal Prism Net
* Five Rectangles: The side faces are 5 identical rectangles. Each is $15 \text{ cm}$ long and $5 \text{ cm}$ wide.
Area = $5 \times (15 \times 5) = 5 \times 75 = 375 \text{ cm}^2$
* Two Pentagons: The problem gives dimensions to split the pentagon into simpler shapes: a central rectangle ($5 \text{ cm}$ wide, $5 \text{ cm}$ high) and two triangles on the sides (base $5 \text{ cm}$, height $2 \text{ cm}$). Note: The label "$7 \text{ cm}$" indicates the total height of the pentagon ($5+2=7$), confirming the triangle height is $2$.
Area of central part = $5 \times 5 = 25 \text{ cm}^2$
Area of two triangles = $2 \times (0.5 \times 5 \times 2) = 10 \text{ cm}^2$
Area of one pentagon = $25 + 10 = 35 \text{ cm}^2$
Area of two pentagons = $2 \times 35 = 70 \text{ cm}^2$
* Total Area: $375 + 70 = \mathbf{445 \text{ cm}^2}$

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Section B: Work out the surface area of the objects

Formula for Cuboids/Rectangular Prisms: $2(lw + lh + wh)$
Formula for Triangular Prisms: $(2 \times \text{Triangle Area}) + (\text{Perimeter of Triangle} \times \text{Length})$

1) Square Prism
* Dimensions: $2 \text{ cm}$, $2 \text{ cm}$, $16 \text{ cm}$
* Ends (Squares): $2 \times (2 \times 2) = 8 \text{ cm}^2$
* Sides (Rectangles): $4 \times (2 \times 16) = 4 \times 32 = 128 \text{ cm}^2$
* Total: $8 + 128 = \mathbf{136 \text{ cm}^2}$

2) Cuboid
* Dimensions: $11 \text{ m}$, $1 \text{ m}$, $7 \text{ m}$
* Pairs of faces:
* $11 \times 1 = 11 \rightarrow \times 2 = 22$
* $11 \times 7 = 77 \rightarrow \times 2 = 154$
* $1 \times 7 = 7 \rightarrow \times 2 = 14$
* Total: $22 + 154 + 14 = \mathbf{190 \text{ m}^2}$

3) Cuboid
* Dimensions: $7 \text{ m}$, $12 \text{ m}$, $4 \text{ m}$
* Pairs of faces:
* $7 \times 12 = 84 \rightarrow \times 2 = 168$
* $7 \times 4 = 28 \rightarrow \times 2 = 56$
* $12 \times 4 = 48 \rightarrow \times 2 = 96$
* Total: $168 + 56 + 96 = \mathbf{320 \text{ m}^2}$

4) Right-Angled Triangular Prism
* Triangle Base: $6 \text{ mm}$, Height: $8 \text{ mm}$ (calculated from hypotenuse $10$ and base $6$, or taken from diagram labels if interpreted as height). Let's use the standard right triangle sides $6, 8, 10$.
* Triangle Area: $0.5 \times 6 \times 8 = 24 \text{ mm}^2$. Two triangles = $48 \text{ mm}^2$.
* Side Rectangles (Length $13 \text{ mm}$):
* $6 \times 13 = 78$
* $8 \times 13 = 104$
* $10 \times 13 = 130$
* Sum of sides = $312 \text{ mm}^2$
* Total: $48 + 312 = \mathbf{360 \text{ mm}^2}$

5) Isosceles Triangular Prism
* Triangle Base: $18 \text{ cm}$, Height: $12 \text{ cm}$. Side lengths: $15 \text{ cm}$.
* Triangle Area: $0.5 \times 18 \times 12 = 108 \text{ cm}^2$. Two triangles = $216 \text{ cm}^2$.
* Side Rectangles (Length $4 \text{ cm}$):
* Perimeter of triangle = $15 + 15 + 18 = 48 \text{ cm}$.
* Area of sides = $48 \times 4 = 192 \text{ cm}^2$.
* Total: $216 + 192 = \mathbf{408 \text{ cm}^2}$

6) Right-Angled Triangular Prism
* Triangle Height: $7 \text{ m}$, Hypotenuse: $17 \text{ m}$.
* Find Base: $\sqrt{17^2 - 7^2} = \sqrt{289 - 49} = \sqrt{240} \approx 15.49 \text{ m}$.
* Triangle Area: $0.5 \times 15.49 \times 7 \approx 54.22 \text{ m}^2$. Two triangles $\approx 108.44 \text{ m}^2$.
* Side Rectangles (Length $9 \text{ m}$):
* Perimeter = $7 + 17 + 15.49 = 39.49 \text{ m}$.
* Area of sides = $39.49 \times 9 \approx 355.41 \text{ m}^2$.
* Total: $108.44 + 355.41 = 463.85$. Rounding to nearest whole number: $\mathbf{464 \text{ m}^2}$

Final Answer:
Section A
1) 68 cm²
2) 216 cm²
3) 445 cm²

Section B
1) 136 cm²
2) 190 m²
3) 320 m²
4) 360 mm²
5) 408 cm²
6) 464 m²