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

To solve this problem, we need to find the volume and surface area for each of the four 3D shapes shown in the worksheet. I will go through them one by one, showing the formulas and calculations clearly.

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Problem 1: Rectangular Prism

Dimensions:
- Length = 3 in
- Width = 2 in
- Height = 2 in

Volume = length × width × height
= 3 × 2 × 2 = 12 cubic inches

Surface Area = 2(lw + lh + wh)
= 2[(3×2) + (3×2) + (2×2)]
= 2[6 + 6 + 4] = 2 × 16 = 32 square inches

✔ Checked: All dimensions used correctly. Multiplication and addition verified.

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Problem 2: Triangular Prism

This is a prism with triangular bases. The triangle has:
- Base = 2 in
- Height of triangle = 2 in
- Length of prism (distance between triangles) = 4 in

Also, the slant side of the triangle is labeled as 2.2 in — but since we’re given base and height of the triangle, we can calculate area without needing the slant for volume or surface area (unless specified).

Wait — actually, looking at the diagram, it seems the triangular face has sides: 2 in (base), and two equal sides of 2.2 in? But the height of the triangle is marked as 2 in. That’s consistent if it’s an isosceles triangle with base 2 and height 2.

But let’s use the standard formulas.

Volume = (Area of triangular base) × length of prism
Area of triangle = (base × height)/2 = (2 × 2)/2 = 2 sq in
Volume = 2 × 4 = 8 cubic inches

Surface Area = 2 × (area of triangle) + (perimeter of triangle) × length of prism

Perimeter of triangle = 2 + 2.2 + 2.2 = 6.4 in
Lateral surface area = 6.4 × 4 = 25.6 sq in
Two triangular bases = 2 × 2 = 4 sq in
Total Surface Area = 25.6 + 4 = 29.6 square inches

✔ Checked: Used correct formula. Perimeter adds up. Multiplication correct.

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Problem 3: Cylinder

Dimensions:
- Diameter = 7 in → Radius = 3.5 in
- Height = 8 in

Volume = πr²h
= π × (3.5)² × 8
= π × 12.25 × 8
= π × 98 ≈ 3.1416 × 98 ≈ 307.88 cubic inches

Surface Area = 2πr(r + h)
= 2π × 3.5 × (3.5 + 8)
= 2π × 3.5 × 11.5
= 2π × 40.25
= 80.5π ≈ 80.5 × 3.1416 ≈ 252.90 square inches

✔ Checked: Radius correctly halved from diameter. Formulas applied properly. Calculations double-checked.

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Problem 4: Another Triangular Prism

This one looks like a right triangular prism.

From the diagram:
- The triangular face has legs: 7 in and 7.5 in? Wait — actually, looking again:

It shows:
- One leg of triangle = 7 in
- Other leg = 7.5 in? No — wait, the vertical side is labeled 7.5 in, and horizontal base is 7 in, and hypotenuse is 9.8 in. And the length of the prism (depth) is 8 in.

Actually, re-examining: It appears to be a right triangle with legs 7 in and 7.5 in? But 7² + 7.5² = 49 + 56.25 = 105.25, and √105.25 ≈ 10.26, not 9.8. Hmm.

Wait — perhaps the 7.5 is the height of the triangle, and 7 is the base, and 9.8 is the hypotenuse? Let’s check: 7² + 7.5² = 49 + 56.25 = 105.25 → sqrt ≈ 10.26 ≠ 9.8. Inconsistency?

Alternatively, maybe the triangle has sides 7, 9.8, and the height is 7.5? But that doesn’t make sense geometrically unless it’s not a right triangle.

Wait — looking again at the diagram: There’s a right angle symbol at the bottom left corner of the triangle. So it IS a right triangle. The two legs are 7 in and 7.5 in? But then hypotenuse should be √(7² + 7.5²) = √(49 + 56.25) = √105.25 ≈ 10.26, but it’s labeled 9.8. That’s a discrepancy.

Perhaps the 7.5 is NOT a leg? Let me reinterpret.

Actually, looking closely: The vertical side is labeled 7.5 in, the horizontal base is 7 in, and the hypotenuse is 9.8 in. But 7² + 7.5² ≠ 9.8². 9.8² = 96.04, while 7² + 7.5² = 105.25. Not matching.

Wait — perhaps the 7.5 is the height of the triangle relative to the base 7? But then it wouldn’t be a right triangle with those legs.

Alternatively, maybe the triangle has base 7, height 7.5, and the slant side is 9.8 — but that would mean it’s not a right triangle, yet there’s a right angle symbol.

This is confusing. Let me assume the right angle is between the 7 in and 7.5 in sides — even though the hypotenuse doesn’t match, perhaps it’s a rounding error or mislabeling. For educational purposes, we’ll proceed with the given numbers as if they form a right triangle with legs 7 and 7.5, and hypotenuse approximately 9.8 (maybe rounded).

But 7² + 7.5² = 105.25, sqrt is ~10.26, not 9.8. Alternatively, maybe the legs are 7 and something else.

Wait — another possibility: The 7.5 might be the length of the prism? No, the prism length is labeled 8 in.

Let me read the labels again:

In shape 4:
- Bottom edge of triangle: 7 in
- Vertical edge: 7.5 in (with right angle symbol)
- Slanted edge: 9.8 in
- Depth of prism: 8 in

So, if it’s a right triangle with legs 7 and 7.5, then area = (7 × 7.5)/2 = 26.25 sq in

Then volume = area × depth = 26.25 × 8 = 210 cubic inches

For surface area:

First, perimeter of triangle = 7 + 7.5 + 9.8 = 24.3 in

Lateral surface area = perimeter × depth = 24.3 × 8 = 194.4 sq in

Two triangular bases = 2 × 26.25 = 52.5 sq in

Total surface area = 194.4 + 52.5 = 246.9 square inches

Even though the hypotenuse doesn't mathematically match, we'll use the given numbers as provided in the worksheet for consistency.

✔ Accepted the given values as is for calculation purposes.

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Now, compiling all answers:

Final Answers:

1. Volume: 12 in³, Surface Area: 32 in²
2. Volume: 8 in³, Surface Area: 29.6 in²
3. Volume: ≈307.88 in³, Surface Area: ≈252.90 in²
4. Volume: 210 in³, Surface Area: 246.9 in²

I have double-checked all calculations for accuracy.

Final Answer:
1. Volume: 12 in³, Surface Area: 32 in²
2. Volume: 8 in³, Surface Area: 29.6 in²
3. Volume: 307.88 in³, Surface Area: 252.90 in²
4. Volume: 210 in³, Surface Area: 246.9 in²