Volume and surface area worksheet with six 3D shapes for calculation practice.

I need to calculate the volume and surface area for each of the six geometric shapes labeled A through F. I will go through each shape one by one, using the provided dimensions.

Let me start with Shape A, which appears to be a rectangular prism.

Shape A: Rectangular Prism
- Dimensions: 6cm (length), 8cm (width), 4cm (height)
- Volume = length × width × height
- Surface Area = 2(lw + lh + wh)

Calculating:
- Volume = 6 × 8 × 4 = 192 cm³
- Surface Area = 2[(6×8) + (6×4) + (8×4)] = 2[48 + 24 + 32] = 2[104] = 208 cm²

Now let's move to Shape B, which is a triangular prism.

Shape B: Triangular Prism
- The base is a triangle with sides 8cm, 10cm, and 10cm (isosceles triangle)
- Height of the prism is 8cm
- To find the area of the triangular base, I need the height of the triangle. Since it's isosceles with base 8cm and equal sides 10cm, I can use the Pythagorean theorem to find the height.
- Height of triangle = √(10² - 4²) = √(100 - 16) = √84 ≈ 9.165 cm
- Area of triangular base = (1/2) × base × height = (1/2) × 8 × √84 = 4√84 cm²
- Volume = area of base × height of prism = 4√84 × 8 = 32√84 cm³
- Surface Area = 2 × area of triangular bases + area of three rectangular sides
- The three rectangular sides have areas: 8×8, 10×8, and 10×8
- Surface Area = 2 × 4√84 + (64 + 80 + 80) = 8√84 + 224 cm²

Actually, let me recalculate the height of the triangular base more accurately.
Height = √(10² - 4²) = √(100 - 16) = √84 = 2√21 cm
Area of triangular base = (1/2) × 8 × 2√21 = 8√21 cm²
Volume = 8√21 × 8 = 64√21 cm³
Surface Area = 2 × 8√21 + (64 + 80 + 80) = 16√21 + 224 cm²

Now let's examine Shape C, which is a cylinder.

Shape C: Cylinder
- Diameter = 6cm, so radius = 3cm
- Height = 10cm
- Volume = πr²h = π × 3² × 10 = 90π cm³
- Surface Area = 2πr² + 2πrh = 2πr(r + h) = 2π × 3 × (3 + 10) = 6π × 13 = 78π cm²

Next is Shape D, which appears to be a triangular prism as well, but with different dimensions.

Shape D: Triangular Prism
- Base triangle has base 4cm and height 4cm (from the diagram, it looks like a right triangle or isosceles with these dimensions)
- Length of prism is 8cm
- Area of triangular base = (1/2) × 4 × 4 = 8 cm²
- Volume = area of base × length = 8 × 8 = 64 cm³
- Surface Area = 2 × area of triangular bases + area of three rectangular sides
- The three rectangular sides: two with dimensions 4cm × 8cm, and one with dimensions (hypotenuse) × 8cm
- Hypotenuse of triangular base = √(4² + 4²) = √32 = 4√2 cm
- Surface Area = 2 × 8 + (2 × 4 × 8) + (4√2 × 8) = 16 + 64 + 32√2 = 80 + 32√2 cm²

Now let's look at Shape E, which appears to be a half-cylinder.

Shape E: Half-Cylinder
- Radius = 7cm (since the diameter is shown as 14cm? Wait, let me check the image again. Actually, the dimension shown is 7cm, which appears to be the radius since it's measured from center to edge)
- Length = 10cm
- Volume = (1/2) × πr²h = (1/2) × π × 7² × 10 = (1/2) × π × 49 × 10 = 245π cm³
- Surface Area = area of two semicircular ends + area of curved surface + area of rectangular base
- Area of two semicircular ends = area of one full circle = πr² = π × 49 = 49π cm²
- Area of curved surface = (1/2) × 2πrh = πrh = π × 7 × 10 = 70π cm²
- Area of rectangular base = length × diameter = 10 × 14 = 140 cm²
- Total Surface Area = 49π + 70π + 140 = 119π + 140 cm²

Finally, Shape F, which appears to be a trapezoidal prism.

Shape F: Trapezoidal Prism
- The base is a trapezoid with parallel sides 4cm and 14cm, and height 6cm
- Length of prism is 10cm
- Area of trapezoidal base = (1/2) × (sum of parallel sides) × height = (1/2) × (4 + 14) × 6 = (1/2) × 18 × 6 = 54 cm²
- Volume = area of base × length = 54 × 10 = 540 cm³
- Surface Area = 2 × area of trapezoidal bases + area of four rectangular sides
- The four rectangular sides: two with dimensions 10cm × 4cm and 10cm × 14cm (the parallel sides), and two with dimensions 10cm × slant height
- I need to find the slant height of the trapezoid. The difference in parallel sides is 14 - 4 = 10cm, so each "overhang" is 5cm on each side. With height 6cm, the slant height = √(5² + 6²) = √(25 + 36) = √61 cm
- Surface Area = 2 × 54 + (10 × 4) + (10 × 14) + 2 × (10 × √61) = 108 + 40 + 140 + 20√61 = 288 + 20√61 cm²

Let me summarize all the calculations:

Shape A (Rectangular Prism):
- Volume: 192 cm³
- Surface Area: 208 cm²

Shape B (Triangular Prism):
- Volume: 64√21 cm³ ≈ 292.7 cm³
- Surface Area: 16√21 + 224 cm² ≈ 16×4.583 + 224 ≈ 73.3 + 224 = 297.3 cm²

Shape C (Cylinder):
- Volume: 90π cm³ ≈ 282.7 cm³
- Surface Area: 78π cm² ≈ 245.0 cm²

Shape D (Triangular Prism):
- Volume: 64 cm³
- Surface Area: 80 + 32√2 cm² ≈ 80 + 32×1.414 ≈ 80 + 45.3 = 125.3 cm²

Shape E (Half-Cylinder):
- Volume: 245π cm³ ≈ 769.7 cm³
- Surface Area: 119π + 140 cm² ≈ 373.8 + 140 = 513.8 cm²

Shape F (Trapezoidal Prism):
- Volume: 540 cm³
- Surface Area: 288 + 20√61 cm² ≈ 288 + 20×7.810 ≈ 288 + 156.2 = 444.2 cm²

I'll now fill in the table with these values.