Grade 6 Geometry Worksheets: Volume and surface area of 3D shapes - Free Printable

Here is the complete solution for each of the 9 problems on the worksheet, calculating the surface area of each prism or cylinder. All answers are rounded to the nearest hundredth where necessary.

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1) Triangular Prism

- Base: Right triangle with legs 2 in and 2 in → hypotenuse = √(2² + 2²) = √8 ≈ 2.828 in
- Height (length) of prism = 4 in

Surface Area = 2 × (Area of triangular base) + (Perimeter of base × height)

- Area of one triangle = (1/2) × 2 × 2 = 2 in²
- Total area of two bases = 2 × 2 = 4 in²
- Perimeter of base = 2 + 2 + √8 ≈ 2 + 2 + 2.828 = 6.828 in
- Lateral surface area = 6.828 × 4 ≈ 27.312 in²
- Total Surface Area = 4 + 27.312 = 31.31 in²

✔ Answer: 31.31 in²

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2) Hexagonal Prism

- Regular hexagon base with side length = 7 mm
- Height of prism = 13 mm

Surface Area = 2 × (Area of hexagon) + (Perimeter × height)

- Area of regular hexagon = (3√3 / 2) × s² = (3√3 / 2) × 49 ≈ (3×1.732/2) × 49 ≈ (5.196/2) × 49 ≈ 2.598 × 49 ≈ 127.302 mm²
- Two bases = 2 × 127.302 = 254.604 mm²
- Perimeter = 6 × 7 = 42 mm
- Lateral area = 42 × 13 = 546 mm²
- Total Surface Area = 254.604 + 546 = 800.60 mm²

✔ Answer: 800.60 mm²

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3) Pentagonal Prism

- Base: Regular pentagon? Not specified — but dimensions given: side = 6 cm, apothem = 8 cm? Wait — image shows height of prism = 14 cm, and base has side = 6 cm, and apothem = 8 cm? Actually, looking at the diagram, it’s a regular pentagonal prism with:

- Side length = 6 cm
- Apothem = 8 cm? That doesn’t match — for a regular pentagon, apothem ≈ 0.688 × side → 6 × 0.688 ≈ 4.13 cm. But here it says “8 cm” — likely the height of the prism is 14 cm, and 8 cm is the apothem? Let’s assume the diagram labels correctly: apothem = 8 cm, side = 6 cm, height = 14 cm.

Actually, rechecking — the diagram shows a pentagon with side 6 cm, and a perpendicular line labeled 8 cm inside — that’s the apothem. So we’ll use that.

Surface Area = 2 × (Area of pentagon) + (Perimeter × height)

- Area of pentagon = (1/2) × perimeter × apothem = (1/2) × (5×6) × 8 = (1/2) × 30 × 8 = 120 cm²
- Two bases = 2 × 120 = 240 cm²
- Perimeter = 5 × 6 = 30 cm
- Lateral area = 30 × 14 = 420 cm²
- Total Surface Area = 240 + 420 = 660.00 cm²

✔ Answer: 660.00 cm²

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4) Cylinder

- Radius = 3 yd
- Height = 4 yd

Surface Area = 2πr² + 2πrh = 2πr(r + h)

- = 2π × 3 × (3 + 4) = 6π × 7 = 42π ≈ 42 × 3.1416 ≈ 131.95 yd²

✔ Answer: 131.95 yd²

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5) Cylinder (same as #4?)

Wait — same diagram? Same radius 3 yd, height 4 yd? Then same answer.

✔ Answer: 131.95 yd²

*(Note: If this was meant to be different, perhaps open top? But diagram shows full cylinder.)*

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6) Triangular Prism (right triangle base)

- Base triangle: legs 3 cm and 4 cm → hypotenuse = 5 cm (Pythagorean triple)
- Height (length) of prism = 11 cm

Surface Area = 2 × (Area of triangle) + (Perimeter × height)

- Area of triangle = (1/2) × 3 × 4 = 6 cm²
- Two bases = 2 × 6 = 12 cm²
- Perimeter = 3 + 4 + 5 = 12 cm
- Lateral area = 12 × 11 = 132 cm²
- Total Surface Area = 12 + 132 = 144.00 cm²

✔ Answer: 144.00 cm²

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7) Cube

- Side = 3 mm

Surface Area = 6 × side² = 6 × 9 = 54 mm²

✔ Answer: 54.00 mm²

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8) Rectangular Prism (Cuboid)

- Dimensions: 11 ft (length), 8 ft (width), 4 ft (height)

Surface Area = 2(lw + lh + wh)

- = 2(11×8 + 11×4 + 8×4) = 2(88 + 44 + 32) = 2(164) = 328 ft²

✔ Answer: 328.00 ft²

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9) Trapezoidal Prism

- Bases of trapezoid: 3 in and 12 in
- Height of trapezoid = 6 in
- Length (height) of prism = 4 in

Surface Area = 2 × (Area of trapezoid) + (Perimeter of trapezoid × length)

First, find area of trapezoid:

- Area = (1/2) × (b₁ + b₂) × h = (1/2) × (3 + 12) × 6 = (1/2) × 15 × 6 = 45 in²
- Two bases = 2 × 45 = 90 in²

Now, perimeter of trapezoid: need all 4 sides.

We have:

- Two parallel sides: 3 in and 12 in
- Height = 6 in → non-parallel sides? The diagram shows slanted sides, but no lengths given. However, the diagram labels one slant side as 4 in? Wait — look again.

Actually, the diagram shows:

- Top base = 3 in
- Bottom base = 12 in
- Height = 6 in
- One leg (slant side) = 4 in? No — wait, the 4 in is the length of the prism, not the side.

Looking carefully — the trapezoid has:

- Bases: 3 in and 12 in
- Height: 6 in
- The non-parallel sides: since the difference in bases is 12 - 3 = 9 in, and if it’s symmetric, each overhang is 4.5 in. Then each slant side = √(6² + 4.5²) = √(36 + 20.25) = √56.25 = 7.5 in

So perimeter = 3 + 12 + 7.5 + 7.5 = 30 in

Lateral surface area = perimeter × length = 30 × 4 = 120 in²

Total surface area = 90 + 120 = 210.00 in²

✔ Answer: 210.00 in²

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## ✔ Final Answers:

1) 31.31 in²
2) 800.60 mm²
3) 660.00 cm²
4) 131.95 yd²
5) 131.95 yd²
6) 144.00 cm²
7) 54.00 mm²
8) 328.00 ft²
9) 210.00 in²

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