This worksheet asks students to calculate the exact surface area for various 3D shapes, including rectangular prisms, triangular prisms, and cylinders.

Here are the step-by-step solutions to find the exact surface area for each shape.

1) Rectangular Prism
* Dimensions: Length = 6 ft, Width = 5 ft, Height = 12 ft.
* Formula: $SA = 2(lw + lh + wh)$
* Calculation:
* Front/Back: $6 \times 12 = 72$
* Top/Bottom: $6 \times 5 = 30$
* Sides: $5 \times 12 = 60$
* Sum: $72 + 30 + 60 = 162$
* Total SA: $162 \times 2 = 324$
* Answer: 324 sq ft

2) Triangular Prism
* Dimensions: Triangle base = 6 yd, height = 4 yd. Side lengths of triangle = 5 yd. Prism length = 11 yd.
* Bases (2 triangles): Area = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 4 = 12$. Two bases = $12 \times 2 = 24$.
* Lateral Faces (3 rectangles): The perimeter of the triangle is $5 + 5 + 6 = 16$.
* Lateral Area = Perimeter $\times$ Length = $16 \times 11 = 176$.
* Total SA: $24 + 176 = 200$.
* Answer: 200 sq yd

3) Rectangular Prism
* Dimensions: Length = 8 in, Width = 14 in, Height = 12 in.
* Formula: $SA = 2(lw + lh + wh)$
* Calculation:
* $8 \times 14 = 112$
* $8 \times 12 = 96$
* $14 \times 12 = 168$
* Sum: $112 + 96 + 168 = 376$
* Total SA: $376 \times 2 = 752$
* Answer: 752 sq in

4) Triangular Prism (Right Triangle)
* Dimensions: Triangle legs = 9 in and 12 in. Hypotenuse = 15 in. Prism length = 9 in.
* Bases (2 triangles): Area = $\frac{1}{2} \times 9 \times 12 = 54$. Two bases = $54 \times 2 = 108$.
* Lateral Faces (3 rectangles): Perimeter of triangle = $9 + 12 + 15 = 36$.
* Lateral Area = $36 \times 9 = 324$.
* Total SA: $108 + 324 = 432$.
* Answer: 432 sq in

5) Cube
* Dimensions: Side = 9 ft.
* Formula: $SA = 6s^2$
* Calculation: $6 \times (9 \times 9) = 6 \times 81 = 486$.
* Answer: 486 sq ft

6) Cylinder
* Dimensions: Radius ($r$) = 10 yd, Height ($h$) = 10 yd.
* Formula: $SA = 2\pi r^2 + 2\pi rh$
* Calculation:
* Bases: $2 \times \pi \times 10^2 = 200\pi$
* Side: $2 \times \pi \times 10 \times 10 = 200\pi$
* Total: $400\pi$
* Answer: $400\pi$ sq yd

7) Cylinder
* Dimensions: Diameter = 2 ft (so Radius $r = 1$ ft), Height ($h$) = 11 ft.
* Formula: $SA = 2\pi r^2 + 2\pi rh$
* Calculation:
* Bases: $2 \times \pi \times 1^2 = 2\pi$
* Side: $2 \times \pi \times 1 \times 11 = 22\pi$
* Total: $2\pi + 22\pi = 24\pi$
* Answer: $24\pi$ sq ft

8) Square Prism
* Dimensions: Base is a square with side 7 m. Height is 14 m.
* Bases (2 squares): Area = $7 \times 7 = 49$. Two bases = $49 \times 2 = 98$.
* Lateral Faces (4 rectangles): Each is $7 \times 14 = 98$. Four faces = $98 \times 4 = 392$.
* Total SA: $98 + 392 = 490$.
* Answer: 490 sq m

9) Triangular Prism
* Dimensions: Triangle base = 10 yd, height = 12 yd. Side lengths = 13 yd. Prism length = 14 yd.
* Bases (2 triangles): Area = $\frac{1}{2} \times 10 \times 12 = 60$. Two bases = $60 \times 2 = 120$.
* Lateral Faces (3 rectangles): Perimeter of triangle = $13 + 13 + 10 = 36$.
* Lateral Area = $36 \times 14 = 504$.
* Total SA: $120 + 504 = 624$.
* Answer: 624 sq yd

Final Answer:
1) 324 sq ft
2) 200 sq yd
3) 752 sq in
4) 432 sq in
5) 486 sq ft
6) $400\pi$ sq yd
7) $24\pi$ sq ft
8) 490 sq m
9) 624 sq yd