Volume Notes

Volume

• Volume is the total cubic units that an object occupies.
• It represents the capacity to fill an object or the three-dimensional space it takes up.

Cubic Centimeter

• A cubic centimeter is a unit of volume, defined as one centimeter by one centimeter by one centimeter.
• 1 \text{ cm} \times 1 \text{ cm} \times 1 \text{ cm}
• Similarly, a cubic foot would be one foot by one foot by one foot.

Volume of a Prism

• The volume of a prism depends on the area of its base.
• The name of a prism is determined by the shape of its base.
• The formula for the volume of any prism is:
  V = \text{Base Area} \times \text{Height of the prism}
• The height must make a right angle with the base.

Example: Triangular Prism

• Consider a triangular prism with a base having dimensions 3 feet by 4 feet, and a height of 10 feet.
• Since it's a right triangle, we don't need the hypotenuse for volume.
• Calculate the base area (triangle) and multiply by the height of the prism.
  • V = (\frac{1}{2} \times 4 \times 3) \times 10
  • V = 6 \times 10 = 60 \text{ cubic feet}

General Prism Considerations

• Prisms have two bases, but the base area is not multiplied by two in the volume formula.
• Multiplying by the height of the prism accounts for the second base, as if layering slices of the base to create the shape.

Volume of a Pyramid

• The volume of a pyramid has a direct ratio with the volume of a prism, if they have the same base and height.
• The ratio is 1:3. It takes three pyramids to fill one prism with the same base and height.
• Formula for the volume of a pyramid:
  V = \frac{1}{3} \times \text{Base Area} \times \text{Altitude Height}

Example: Rectangular Pyramid

• Consider a rectangular pyramid with a base of 10 by 6 and a height of 12.
• V = \frac{1}{3} \times (6 \times 10) \times 12
• V = \frac{1}{3} \times 60 \times 12 = 240 \text{ cubic units}

Volume of a Cylinder

• When filling a cylinder with water, the water spreads out to cover the base area first and then rises.
• The formula for the volume of a cylinder is:
  V = \text{Base Area} \times \text{Height of the cylinder}
• More specifically, since the base of a cylinder is always a circle:
  V = \pi r^2 h

Example: Cylinder

• Given a cylinder with a radius of 3 meters and a height of 4 meters:
  V = 3.14 \times 3^2 \times 4
  V = 3.14 \times 9 \times 4 = 113.04 \text{ cubic meters}

Volume of a Cone

• A cone's volume also has a 1:3 ratio with a cylinder, assuming they have the same base area and height.
• The formula for the volume of a cone is:
  V = \frac{1}{3} \pi r^2 h
• Where h is the altitude height.
• Slant height is not needed for volume calculation unless the altitude height is unknown and needs to be calculated using the Pythagorean theorem.

Example: Cone

• Given a cone with a radius of 7 and a height of 10:
  V = \frac{1}{3} \times 3.14 \times 7^2 \times 10
  V \approx 512.87 \text{ cubic units}

Volume of a Sphere

• The formula for the volume of a sphere is:
  V = \frac{4}{3} \pi r^3
• The only variable needed is the radius.

Example: Sphere

• Given a sphere with a diameter of 8 (so radius = 4), express the volume in terms of \pi:
  V = \frac{4}{3} \pi (4)^3
  V = \frac{4}{3} \pi (64)
  V = \frac{256}{3} \pi \text{ cubic units}
  V \approx 85.33 \pi \text{ cubic units}

Composite Shapes

• For composite shapes, you either add or subtract volumes.

Adding Volumes

• Example: Snow cone (hemisphere + cone).
• \text{Volume} = \text{Volume of Hemisphere} + \text{Volume of Cone}
• V = (\frac{2}{3} \pi r^3) + (\frac{1}{3} \pi r^2 h)

Subtracting Volumes

• If a shape is within a shape, subtract the volumes.

• Example: Tennis balls in a cylinder.

• \text{Volume of air} = \text{Volume of Cylinder} - 3 \times \text{Volume of Sphere}

• V = (\pi r^2 h) - 3 \times (\frac{4}{3} \pi r^3)

• Overlapping or exposed faces are not a concern when calculating volume.