Volume and Surface Area of 3D Shapes Worksheet - Practice calculating volume and surface area for various geometric solids.

Here are the step-by-step solutions for each shape on the worksheet.

Row 1

1. Rectangular Prism (Top Left)
* Dimensions: Length = 10 mm, Width = 6 mm, Height = 3 mm.
* Volume: Multiply length × width × height.
$$10 \times 6 \times 3 = 180$$
* Surface Area: Add the area of all 6 sides ($2lw + 2lh + 2wh$).
* Top/Bottom: $10 \times 6 = 60$
* Front/Back: $10 \times 3 = 30$
* Sides: $6 \times 3 = 18$
* Total: $2(60 + 30 + 18) = 2(108) = 216$

2. Cube (Top Middle-Left)
* Dimensions: Side = 4 mm.
* Volume: Side × Side × Side ($s^3$).
$$4 \times 4 \times 4 = 64$$
* Surface Area: 6 identical square faces ($6s^2$).
$$6 \times (4 \times 4) = 6 \times 16 = 96$$

3. Cylinder (Top Middle-Right)
* Dimensions: Radius ($r$) = 7 cm, Height ($h$) = 10 cm.
* Volume: $\pi r^2 h$.
$$\pi \times 7^2 \times 10 = \pi \times 49 \times 10 = 490\pi$$
*(Using $\pi \approx 3.14$: $490 \times 3.14 = 1538.6$)*
* Surface Area: $2\pi r^2 + 2\pi rh$.
* Two circles (top/bottom): $2 \times \pi \times 7^2 = 98\pi$
* Curved side: $2 \times \pi \times 7 \times 10 = 140\pi$
* Total: $98\pi + 140\pi = 238\pi$
*(Using $\pi \approx 3.14$: $238 \times 3.14 = 747.32$)*

4. Triangular Prism (Top Right)
* Dimensions: Triangle base = 15 cm, Triangle height = 8 cm, Prism length = 10 cm. The slanted side is 17 cm.
* Volume: Area of triangle base × Prism length.
* Triangle Area: $\frac{1}{2} \times 15 \times 8 = 60$
* Volume: $60 \times 10 = 600$
* Surface Area: 2 Triangles + 3 Rectangles.
* 2 Triangles: $2 \times 60 = 120$
* Bottom Rectangle: $15 \times 10 = 150$
* Back Rectangle: $8 \times 10 = 80$
* Slanted Rectangle: $17 \times 10 = 170$
* Total: $120 + 150 + 80 + 170 = 520$

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Row 2

5. Square Pyramid (Bottom Left)
* Dimensions: Base side = 4 yd, Slant height = 10 yd.
* Surface Area: Base Area + Lateral Area (4 triangles).
* Base Area: $4 \times 4 = 16$
* One Triangle Area: $\frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 4 \times 10 = 20$
* 4 Triangles: $4 \times 20 = 80$
* Total Surface Area: $16 + 80 = 96$
*(Note: Volume cannot be calculated exactly because the vertical height is missing).*

6. Rectangular Pyramid (Bottom Middle-Left)
* Dimensions: Base = 8 cm by 3 cm, Vertical Height ($h$) = 10 cm.
* Volume: $\frac{1}{3} \times \text{Base Area} \times \text{Height}$.
* Base Area: $8 \times 3 = 24$
* Volume: $\frac{1}{3} \times 24 \times 10 = 8 \times 10 = 80$

7. Triangular Prism (Bottom Middle-Right)
* Dimensions: Triangle base = 8 m, Triangle height = 3 m, Prism length = 6 m. The slanted side is 5 m.
* Volume: Area of triangle base × Prism length.
* Triangle Area: $\frac{1}{2} \times 8 \times 3 = 12$
* Volume: $12 \times 6 = 72$
* Surface Area: 2 Triangles + 3 Rectangles.
* 2 Triangles: $2 \times 12 = 24$
* Bottom Rectangle: $8 \times 6 = 48$
* Back Rectangle: $3 \times 6 = 18$
* Slanted Rectangle: $5 \times 6 = 30$
* Total: $24 + 48 + 18 + 30 = 120$

8. Cylinder (Bottom Right)
* Dimensions: Diameter = 10 cm (so Radius $r = 5$ cm), Height = 8 cm.
* Volume: $\pi r^2 h$.
$$\pi \times 5^2 \times 8 = \pi \times 25 \times 8 = 200\pi$$
*(Using $\pi \approx 3.14$: $200 \times 3.14 = 628$)*
* Surface Area: $2\pi r^2 + 2\pi rh$.
* Two circles: $2 \times \pi \times 5^2 = 50\pi$
* Curved side: $2 \times \pi \times 5 \times 8 = 80\pi$
* Total: $50\pi + 80\pi = 130\pi$
*(Using $\pi \approx 3.14$: $130 \times 3.14 = 408.2$)*

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Final Answer:

Row 1:
1. Rectangular Prism: Volume = 180 mm³, Surface Area = 216 mm²
2. Cube: Volume = 64 mm³, Surface Area = 96 mm²
3. Cylinder: Volume = 490$\pi$ cm³ (approx. 1538.6), Surface Area = 238$\pi$ cm² (approx. 747.32)
4. Triangular Prism: Volume = 600 cm³, Surface Area = 520 cm²

Row 2:
5. Square Pyramid: Surface Area = 96 yd²
6. Rectangular Pyramid: Volume = 80 cm³
7. Triangular Prism: Volume = 72 m³, Surface Area = 120 m²
8. Cylinder: Volume = 200$\pi$ cm³ (approx. 628), Surface Area = 130$\pi$ cm² (approx. 408.2)