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Volume Calculations of 3D Solids

Master the formulas for calculating volumes of common three-dimensional shapes

Interactive Volume Calculators

Cylinder

$V = \pi r^2 h$

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Cone

$V = \frac{1}{3}\pi r^2 h$

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Pyramid

$V = \frac{1}{3} \times \text{Base Area} \times h$

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Prism

$V = \text{Base Area} \times h$

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Sphere

$V = \frac{4}{3}\pi r^3$

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Volume Formulas for 3D Solids

Understanding how to calculate the volume of three-dimensional shapes is essential for solving problems in geometry, physics, and engineering. Each type of solid has a specific formula based on its geometric properties.

1. Cylinders

A cylinder is formed by rotating a rectangle around one of its sides. The volume represents the space inside this circular solid.

Cylinder Volume Formulas:

$$V = \pi r^2 h$$
$$V = B \times h \text{ (where B is the base area)}$$

where r is the radius and h is the height

The volume can be conceptualized as stacking infinite circular layers (disks) of area πr² to a height h.

Example: Cylinder Volume

Problem: Find the volume of a cylindrical tank with radius 3 meters and height 8 meters.

Solution: $V = \pi r^2 h = \pi \times 3^2 \times 8 = \pi \times 9 \times 8 = 72\pi \approx 226.19 \text{ m}^3$

2. Cones

A cone is formed by rotating a right triangle around one of its legs. The volume of a cone is exactly one-third the volume of a cylinder with the same base and height.

Cone Volume Formulas:

$$V = \frac{1}{3}\pi r^2 h$$
$$V = \frac{1}{3} \times \text{Base Area} \times h$$

where r is the radius of the base and h is the height

Example: Cone Volume

Problem: A cone has a base radius of 4 cm and height of 9 cm. What is its volume?

Solution: $V = \frac{1}{3}\pi r^2 h = \frac{1}{3} \times \pi \times 4^2 \times 9 = \frac{1}{3} \times \pi \times 16 \times 9 = 48\pi \approx 150.8 \text{ cm}^3$

3. Pyramids

A pyramid has a polygonal base and triangular faces that meet at a point (apex). Like cones, the volume of a pyramid is one-third the volume of a prism with the same base and height.

Pyramid Volume Formula:

$$V = \frac{1}{3} \times \text{Base Area} \times h$$

For a square pyramid with side length s: Base Area = s², so $V = \frac{1}{3}s^2 h$

Example: Square Pyramid Volume

Problem: A square pyramid has a base with side length 6 cm and height 10 cm.

Solution: $\text{Base Area} = 6^2 = 36 \text{ cm}^2$

$V = \frac{1}{3} \times 36 \times 10 = 120 \text{ cm}^3$

4. Prisms

A prism has two parallel, congruent bases connected by rectangular faces. The volume is simply the base area multiplied by the height.

Prism Volume Formula:

$$V = \text{Base Area} \times h$$

This works for any prism: triangular, rectangular, pentagonal, etc.

Example: Triangular Prism Volume

Problem: A triangular prism has a base area of 15 cm² and height of 12 cm.

Solution: $V = \text{Base Area} \times h = 15 \times 12 = 180 \text{ cm}^3$

5. Spheres

A sphere is formed by rotating a semicircle around its diameter. It's a perfectly round three-dimensional object.

Sphere Volume Formula:

$$V = \frac{4}{3}\pi r^3$$

where r is the radius of the sphere

Example: Sphere Volume

Problem: Find the volume of a sphere with radius 5 cm.

Solution: $V = \frac{4}{3}\pi r^3 = \frac{4}{3} \times \pi \times 5^3 = \frac{4}{3} \times \pi \times 125 = \frac{500}{3}\pi \approx 523.6 \text{ cm}^3$

Key Relationships to Remember

  • Cone vs. Cylinder: A cone's volume is exactly $\frac{1}{3}$ of a cylinder with the same base and height
  • Pyramid vs. Prism: A pyramid's volume is exactly $\frac{1}{3}$ of a prism with the same base and height
  • Base Area: For circular bases, B = πr²; for square bases, B = s²
  • Units: Volume is always measured in cubic units (cm³, m³, etc.)

Problem-Solving Strategy

  1. Identify the shape: Determine which type of 3D solid you're working with
  2. Find the appropriate formula: Use the correct volume formula for that shape
  3. Identify given values: Note which measurements are provided
  4. Calculate missing values: Find any needed measurements (like base area)
  5. Substitute and solve: Plug values into the formula and calculate
  6. Check units: Ensure your answer is in the correct cubic units

Composite Solids

Sometimes you'll encounter complex shapes made from combining or subtracting basic solids. For these:

  • Addition: Add volumes when shapes are joined together
  • Subtraction: Subtract volumes when shapes are removed (like hollow objects)
  • Break down: Decompose complex shapes into simpler ones

Practice Problem

Problem:

A cylindrical water tank has a radius of 2 meters and height of 6 meters. A conical cap with the same radius sits on top, adding 1.5 meters to the total height. What is the total volume?

Show Solution

Step 1: Calculate cylinder volume

$V_{\text{cylinder}} = \pi r^2 h = \pi \times 2^2 \times 6 = 24\pi \text{ m}^3$

Step 2: Calculate cone volume

$V_{\text{cone}} = \frac{1}{3}\pi r^2 h = \frac{1}{3} \times \pi \times 2^2 \times 1.5 = 2\pi \text{ m}^3$

Step 3: Add volumes

$\text{Total Volume} = 24\pi + 2\pi = 26\pi \approx 81.68 \text{ m}^3$

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