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

✔ Check: All faces are same, 6 faces → correct.

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Problem 2: Triangular Prism

This prism has:
- Triangle base with base = 2 in, height = 2 in
- Length of prism = 8 in
- The two triangular ends are identical.
- The three rectangular sides: one is 8×2 (bottom), and two are slanted — but we’re given the slant edge as 2.2 in? Wait — actually, looking at the diagram, it shows:

The triangle has:
- Base = 2 in
- Height = 2 in
- Two equal sides? Actually, the diagram labels the two slanted edges as 2.2 in each — so it’s an isosceles triangle with base 2 in, height 2 in, and legs 2.2 in.

But wait — if base is 2 and height is 2, then using Pythagoras, half-base is 1, so leg should be √(1² + 2²) = √5 ≈ 2.236 — close to 2.2, so they rounded it. We’ll use 2.2 as given.

So:

Volume of triangular prism = (Area of triangle base) × length
Area of triangle = (base × height)/2 = (2 × 2)/2 = 2 sq in
Volume = 2 × 8 = 16 cubic inches

Surface Area = sum of all faces

Faces:
- Two triangles: 2 × 2 = 4 sq in
- Three rectangles:
- Bottom rectangle: 8 in × 2 in = 16 sq in
- Two side rectangles: each is 8 in × 2.2 in = 17.6 sq in each → total 35.2 sq in

Total surface area = 4 + 16 + 35.2 = 55.2 square inches

✔ Check: Makes sense — triangles small, rectangles larger.

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Problem 3: Cylinder

Given:
- Radius = 7 in (since diameter is shown as 7 in? Wait — look again.)

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

So radius r = 7 ÷ 2 = 3.5 in

Height h = 8 in

Volume of cylinder = π × r² × h
= π × (3.5)² × 8
= π × 12.25 × 8
= π × 98
≈ 3.14 × 98 = 307.72 cubic inches

Surface Area of cylinder = 2πr² + 2πrh
= 2πr(r + h)
= 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 × 3.14 = 252.77 square inches

✔ Check: Formula applied correctly. Used diameter to get radius.

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Problem 4: Right Triangular Prism (with right triangle base)

Base is a right triangle:
- Legs = 7 in and 7 in (it’s a right isosceles triangle)
- Hypotenuse = 9.9 in (given — which makes sense: √(7²+7²)=√98≈9.899≈9.9)
- Length of prism = 8 in

Volume = (Area of triangle base) × length
Area of triangle = (7 × 7)/2 = 49/2 = 24.5 sq in
Volume = 24.5 × 8 = 196 cubic inches

Surface Area = sum of all faces

Faces:
- Two triangles: 2 × 24.5 = 49 sq in
- Three rectangles:
- Rectangle on leg 7 in: 7 × 8 = 56 sq in
- Rectangle on other leg 7 in: 7 × 8 = 56 sq in
- Rectangle on hypotenuse 9.9 in: 9.9 × 8 = 79.2 sq in

Total surface area = 49 + 56 + 56 + 79.2 =
49 + 112 = 161; 161 + 79.2 = 240.2 square inches

✔ Check: All faces accounted for. Calculations add up.

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Final Answer:

1. Volume: 27 in³, Surface Area: 54 in²
2. Volume: 16 in³, Surface Area: 55.2 in²
3. Volume: 307.72 in³, Surface Area: 252.77 in²
4. Volume: 196 in³, Surface Area: 240.2 in²