Volume and Surface Area Practice | PDF | Volume | Area - Free Printable

Let me work through each problem step by step.

Problem 1: Cube with side 9 cm
- Surface Area = 6s² = 6(9)² = 6(81) = 486 cm²
- Volume = s³ = 9³ = 729 cm³

Problem 2: Rectangular Prism (11 mm × 3 mm × 5 mm)
- Surface Area = 2(lw + lh + wh) = 2(11×3 + 11×5 + 3×5) = 2(33 + 55 + 15) = 2(103) = 206 mm²
- Volume = lwh = 11 × 3 × 5 = 165 mm³

Problem 3: Triangular Prism
The triangular base has legs 16 in and 30 in, and the prism length is 9 in.
- The triangle is a right triangle with legs 16 and 30.
- Hypotenuse = √(16² + 30²) = √(256 + 900) = √1156 = 34 in
- Area of triangular base = (1/2)(16)(30) = 240 in²
- Perimeter of triangle = 16 + 30 + 34 = 80 in
- Surface Area = 2(base area) + perimeter × length = 2(240) + 80(9) = 480 + 720 = 1200 in²
- Volume = base area × length = 240 × 9 = 2160 in³

Problem 4: Triangular Prism
The triangular base has legs 10 ft and... let me look again. It shows 10 ft on one leg, 6 ft on another dimension, and 14.5 ft appears to be the hypotenuse or the length of the prism.

Looking more carefully: The right triangle has legs 10 ft and some other value, with 14.5 ft shown as what appears to be the hypotenuse of the triangle or the prism length. The 6 ft appears to be the length of the prism.

Wait, let me reconsider. Looking at the diagram: 10 ft is one leg of the right triangle, 6 ft appears to be the prism length (the depth), and 14.5 ft is likely the hypotenuse of the triangular face.

If 10 ft is one leg and 14.5 ft is the hypotenuse, then the other leg = √(14.5² - 10²) = √(210.25 - 100) = √110.25 = 10.5 ft

Hmm, but that seems odd. Let me reconsider.

Actually, looking at typical problems like this, 14.5 ft might be the slant edge (hypotenuse of the triangular face), 10 ft is one leg, and we need to find the other leg. Or 14.5 could be the prism length.

Let me try: if the triangle has legs 10 and some value, and 14.5 is the hypotenuse:
Other leg = √(14.5² - 10²) = √(210.25 - 100) = √110.25 = 10.5 ft

Then:
- Area of triangle = (1/2)(10)(10.5) = 52.5 ft²
- Perimeter = 10 + 10.5 + 14.5 = 35 ft
- If 6 ft is the prism length:
- Surface Area = 2(52.5) + 35(6) = 105 + 210 = 315 ft²
- Volume = 52.5 × 6 = 315 ft³

Actually wait, let me reconsider the diagram. The 14.5 ft might actually be the length of the prism (the depth), and the triangle has legs 10 ft and 6 ft.

If triangle legs are 10 and 6:
- Hypotenuse = (100 + 36) = 136 ≈ 11.66 ft
- Area = (1/2)(10)(6) = 30 ft²
- Perimeter = 10 + 6 + 11.66 = 27.66 ft
- If 14.5 is prism length:
- SA = 2(30) + 27.66(14.5) = 60 + 401.07 = 461.07 ft²
- Volume = 30 × 14.5 = 435 ft³

Hmm, let me think about this differently. In the diagram for problem 4, it looks like:
- 10 ft is the vertical leg of the right triangle
- 6 ft is the horizontal leg (or the prism length going into the page)
- 14.5 ft is shown inside/near the triangle

Given the way these problems are typically set up, I think 14.5 ft is the hypotenuse of the triangular face, 10 ft is one leg, and we need to find the other leg. Then 6 ft is the prism length.

Other leg = √(14.5² - 10²) = √(210.25 - 100) = √110.25 = 10.5 ft

- Area of triangle = (1/2)(10)(10.5) = 52.5 ft²
- Perimeter = 10 + 10.5 + 14.5 = 35 ft
- Prism length = 6 ft
- Surface Area = 2(52.5) + 35(6) = 105 + 210 = 315 ft²
- Volume = 52.5 × 6 = 315 ft³

Problem 5: Rectangular prism with SA = 448 cm², length = 14 cm, width = 6 cm, find height
SA = 2(lw + lh + wh)
448 = 2(14×6 + 14h + 6h)
448 = 2(84 + 20h)
224 = 84 + 20h
140 = 20h
h = 7 cm

Problem 6: Cylinder with r = 12 cm, h = 15 cm
- Surface Area = 2πr² + 2πrh = 2π(144) + 2π(12)(15) = 288π + 360π = 648π cm² ≈ 2035.75 cm²
- Volume = πr²h = π(144)(15) = 2160π cm³ ≈ 6785.84 cm³

Problem 7: Cylinder with diameter = 10 in (so r = 5 in), h = 5 in
- Surface Area = 2πr² + 2πrh = 2π(25) + 2π(5)(5) = 50π + 50π = 100π in² ≈ 314.16 in²
- Volume = πr²h = π(25)(5) = 125π in³ ≈ 392.70 in³

Problem 8: Cylinder with r = 7.5 mm, h = 12.5 mm
- Surface Area = 2πr² + 2πrh = 2π(56.25) + 2π(7.5)(12.5) = 112.5π + 187.5π = 300π mm² ≈ 942.48 mm²
- Volume = πr²h = π(56.25)(12.5) = 703.125π mm³ ≈ 2208.93 mm³

Let me verify my calculations using code.