Volume and surface area of 3D shapes worksheet for Grade 6 geometry practice.

I'll solve the volume and surface area problems for each of the 3D shapes shown in this Grade 6 Geometry Worksheet.

Let me work through each shape one by one:

## Shape 1: Cube
- Dimensions: 3 in × 3 in × 3 in

Volume: V = side³ = 3³ = 27 cubic inches
Surface Area: SA = 6 × side² = 6 × 3² = 6 × 9 = 54 square inches

## Shape 2: Triangular Prism
- Base triangle dimensions: base = 2 in, height = 2.2 in
- Length of prism = 8 in

Volume: V = (base area) × length = (½ × 2 × 2.2) × 8 = (2.2) × 8 = 17.6 cubic inches
Surface Area: SA = 2 × (triangle area) + 3 × (rectangular faces)
- Triangle area = ½ × 2 × 2.2 = 2.2 in²
- Two triangular faces = 2 × 2.2 = 4.4 in²
- Three rectangular faces:
- Two with dimensions 2 in × 8 in = 2 × (2 × 8) = 32 in²
- One with dimensions 2.2 in × 8 in = 17.6 in²
- Total SA = 4.4 + 32 + 17.6 = 54 in²

Wait, let me reconsider the surface area calculation for the triangular prism. The three rectangular faces should be:
- Two rectangles with dimensions 2 in × 8 in = 2 × (2 × 8) = 32 in²
- One rectangle with dimensions equal to the hypotenuse of the triangle times 8 in. But since we don't have the hypotenuse, and the diagram shows a right triangle, I need to recalculate.

Actually, looking at the diagram again, it appears to be a right triangular prism where the sides are 2 in, 2.2 in, and the hypotenuse. But for surface area, we need all three rectangular faces.

Let me calculate the hypotenuse using Pythagorean theorem: √(2² + 2.2²) = √(4 + 4.84) = √8.84 ≈ 2.97 in

So the three rectangular faces are:
- 2 in × 8 in = 16 in²
- 2.2 in × 8 in = 17.6 in²
- 2.97 in × 8 in ≈ 23.76 in²

Total SA = 4.4 + 16 + 17.6 + 23.76 = 61.76 in²

This seems overly complex for a Grade 6 worksheet. Let me reconsider - perhaps the diagram is showing the dimensions differently.

Looking again at the diagram, it might be that the 2.2 in is the height of the triangle, and the 2 in is the base, and the length is 8 in. For surface area, we have:
- Two triangular bases: 2 × (½ × 2 × 2.2) = 4.4 in²
- Three rectangular sides:
- One with dimensions 2 in × 8 in = 16 in²
- One with dimensions 2.2 in × 8 in = 17.6 in²
- One with dimensions (the third side of the triangle) × 8 in

Since we don't have the third side of the triangle, and given this is a Grade 6 worksheet, I suspect there might be an error in my interpretation or the problem might be simplified.

Let me move on to the other shapes and come back if needed.

## Shape 3: Cylinder
- Diameter = 7 in, so radius = 3.5 in
- Height = 8 in

Volume: V = π × r² × h = π × (3.5)² × 8 = π × 12.25 × 8 = 98π ≈ 307.88 cubic inches
Surface Area: SA = 2πr² + 2πrh = 2πr(r + h) = 2π × 3.5 × (3.5 + 8) = 7π × 11.5 = 80.5π ≈ 252.86 square inches

## Shape 4: Triangular Prism (Right Triangular Prism)
- Base triangle: base = 7 in, height = 7 in (right triangle)
- Length of prism = 8 in

Volume: V = (base area) × length = (½ × 7 × 7) × 8 = (24.5) × 8 = 196 cubic inches
Surface Area: SA = 2 × (triangle area) + 3 × (rectangular faces)
- Triangle area = ½ × 7 × 7 = 24.5 in²
- Two triangular faces = 2 × 24.5 = 49 in²
- Three rectangular faces:
- One with dimensions 7 in × 8 in = 56 in²
- One with dimensions 7 in × 8 in = 56 in²
- One with dimensions (hypotenuse) × 8 in
- Hypotenuse = √(7² + 7²) = √98 = 7√2 ≈ 9.9 in
- So this face ≈ 9.9 × 8 = 79.2 in²
- Total SA = 49 + 56 + 56 + 79.2 = 240.2 in²

Again, this seems complex for Grade 6. Let me reconsider the surface area calculation for Shape 2.

For Shape 2, if we assume it's a right triangular prism with legs 2 in and 2.2 in, and length 8 in:
- Volume: 17.6 in³ (as calculated)
- Surface Area: 2 × (½ × 2 × 2.2) + (2 × 8) + (2.2 × 8) + (√(2² + 2.2²) × 8)
- = 4.4 + 16 + 17.6 + (√8.84 × 8)
- = 4.4 + 16 + 17.6 + (2.97 × 8)
- = 4.4 + 16 + 17.6 + 23.76
- = 61.76 in²

Given the complexity, I'll provide the most straightforward calculations for each shape:

## Final Answers:

Shape 1 (Cube):
- Volume: 27 in³
- Surface Area: 54 in²

Shape 2 (Triangular Prism):
- Volume: 17.6 in³
- Surface Area: Approximately 61.76 in² (or if simplified, possibly 54 in² if only considering visible faces, but this would be incorrect)

Shape 3 (Cylinder):
- Volume: 98π in³ ≈ 307.88 in³
- Surface Area: 80.5π in² ≈ 252.86 in²

Shape 4 (Triangular Prism):
- Volume: 196 in³
- Surface Area: Approximately 240.2 in²

Note: For educational purposes at the Grade 6 level, the surface area calculations for the triangular prisms might be simplified or the diagrams might have intended different interpretations. The volume calculations are straightforward and correct.