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

Let’s solve each problem one by one. We’ll find both the volume and surface area for each 3D shape.

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Problem 1: Cube (all sides = 3 in)

Volume of a cube = side × side × side
= 3 × 3 × 3 = 27 cubic inches

Surface area of a cube = 6 × (side × side)
= 6 × (3 × 3) = 6 × 9 = 54 square inches

✔ Volume = 27 in³, Surface Area = 54 in²

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Problem 2: Triangular Prism

This prism has:
- A triangular base with base = 2 in, height = 2.2 in
- Length of prism = 8 in
- The other two sides of the triangle are also 2.2 in (it’s an isosceles triangle)

Step 1: Volume

Volume of any prism = Area of base × length

Area of triangular base = (base × height) ÷ 2
= (2 × 2.2) ÷ 2 = 4.4 ÷ 2 = 2.2 in²

Volume = 2.2 × 8 = 17.6 in³

Step 2: Surface Area

We need to add up all the faces:

- Two triangular bases: 2 × 2.2 = 4.4 in²
- Three rectangular sides:
- Bottom rectangle: 2 in × 8 in = 16 in²
- Two slanted rectangles: each is 2.2 in × 8 in → 2 × (2.2 × 8) = 2 × 17.6 = 35.2 in²

Total surface area = 4.4 + 16 + 35.2 = 55.6 in²

✔ Volume = 17.6 in³, Surface Area = 55.6 in²

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Problem 3: Cylinder

Given:
- Radius = 7 in (since diameter is shown as 7? Wait — let’s check!)

Wait! In the diagram, it says “7 in” across the bottom — that’s the diameter, not radius.

So radius = 7 ÷ 2 = 3.5 in

Height = 8 in

Volume of cylinder = π × r² × h
≈ 3.14 × (3.5)² × 8
First, 3.5² = 12.25
Then, 3.14 × 12.25 = 38.465
Then, 38.465 × 8 = 307.72 in³ (we can round to nearest tenth if needed → 307.7 in³)

But let’s keep more precision for now.

Actually, let’s use exact steps:

π × 3.5² × 8 = π × 12.25 × 8 = π × 98 ≈ 3.1416 × 98 ≈ 307.8768 in³ → we’ll say 307.9 in³

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

Plug in:

2 × π × 3.5 × (3.5 + 8) = 2 × π × 3.5 × 11.5

First, 3.5 × 11.5 = 40.25
Then, 2 × 40.25 = 80.5
Then, 80.5 × π ≈ 80.5 × 3.1416 ≈ 252.9 in²

Alternatively, calculate separately:

Top and bottom circles: 2 × π × r² = 2 × π × 12.25 = 24.5π ≈ 76.97 in²

Side (lateral): 2πrh = 2 × π × 3.5 × 8 = 56π ≈ 175.93 in²

Total = 76.97 + 175.93 = 252.9 in²

✔ Volume ≈ 307.9 in³, Surface Area ≈ 252.9 in²

*(Note: If your teacher uses π = 3.14 exactly, then:*

*Volume = 3.14 × 12.25 × 8 = 3.14 × 98 = 307.72 → 307.7 in³*

*Surface Area = 2×3.14×3.5×(3.5+8) = 2×3.14×3.5×11.5 = 6.28×40.25 = 252.77 → 252.8 in²)*

We’ll go with π = 3.14 for consistency unless told otherwise.

So final for #3 using π = 3.14:

Volume = 3.14 × 3.5² × 8 = 3.14 × 12.25 × 8 = 3.14 × 98 = 307.72 → 307.7 in³

Surface Area = 2×3.14×3.5×3.5 + 2×3.14×3.5×8
= 2×3.14×12.25 + 2×3.14×28
= 76.93 + 175.84 = 252.77 → 252.8 in²

✔ Let’s use: Volume = 307.7 in³, Surface Area = 252.8 in²

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Problem 4: Right Triangular Prism (right triangle base)

Base is a right triangle with legs = 7 in and 7 in (wait — look again!)

Diagram shows:
- One leg = 7 in (horizontal)
- Other leg = 7 in (vertical) — marked with right angle
- Hypotenuse = 9.9 in (given)
- Length of prism = 8 in

Actually, wait — the vertical side is labeled 7 in, horizontal base is 7 in, hypotenuse 9.9 in, and depth (length of prism) is 8 in.

Yes.

Volume = Area of triangular base × length

Area of triangle = (base × height) ÷ 2 = (7 × 7) ÷ 2 = 49 ÷ 2 = 24.5 in²

Volume = 24.5 × 8 = 196 in³

Surface Area:

Two triangular bases: 2 × 24.5 = 49 in²

Three rectangular sides:

- Rectangle on base (7 in × 8 in) = 56 in²
- Rectangle on height (7 in × 8 in) = 56 in²
- Rectangle on hypotenuse (9.9 in × 8 in) = 79.2 in²

Total surface area = 49 + 56 + 56 + 79.2 = 240.2 in²

✔ Volume = 196 in³, Surface Area = 240.2 in²

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## Final Answers:

Problem 1:
Volume = 27 in³, Surface Area = 54 in²

Problem 2:
Volume = 17.6 in³, Surface Area = 55.6 in²

Problem 3:
Volume = 307.7 in³, Surface Area = 252.8 in²

Problem 4:
Volume = 196 in³, Surface Area = 240.2 in²

Final Answer:
1. Volume: 27 in³, Surface Area: 54 in²
2. Volume: 17.6 in³, Surface Area: 55.6 in²
3. Volume: 307.7 in³, Surface Area: 252.8 in²
4. Volume: 196 in³, Surface Area: 240.2 in²