Volume of cylinders, cones, spheres Knowledge-Based Study

Study notes

1. Understanding Cylindrical Volume

   A cylinder is a three-dimensional solid with two parallel, congruent circular bases connected by a curved lateral surface. Because the cross-section parallel to the bases is uniform throughout its height, the volume of a cylinder is found by multiplying the area of its circular base by its perpendicular height.

   The area of a circular base with radius $r$ is given by $\pi r^2$. Therefore, the formula for the volume $V$ of a cylinder with height $h$ is:

   $$V = \pi r^2 h$$

   Worked Example

   Find the volume of a cylinder with a radius of 4 cm and a height of 10 cm. Express your answer in terms of $\pi$ and as an approximation using $\pi \approx 3.14$.

   • Base area: $B = \pi (4)^2 = 16\pi\text{ cm}^2$
   • Volume: $V = 16\pi \times 10 = 160\pi\text{ cm}^3$
   • Decimal approximation: $V \approx 160 \times 3.14 = 502.4\text{ cm}^3$

   Key Tip: Always verify whether a given measurement represents the radius or the diameter. If the diameter is given, divide it by 2 before substituting into the formula.

   Relating Cones to Cylinders

   A cone has a single circular base that tapers smoothly to a point called the apex. When a cone and a cylinder share the exact same base radius $r$ and vertical height $h$, the volume of the cone is precisely one-third the volume of the cylinder.

   The formula for the volume $V$ of a cone is:

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

   Height versus Slant Height

   A frequent error is substituting the slant height $l$ instead of the perpendicular height $h$. The perpendicular height forms a right angle with the radius at the center of the base. If only the slant height is provided, use the Pythagorean theorem to calculate the vertical height:

   $$r^2 + h^2 = l^2 \implies h = \sqrt{l^2 - r^2}$$

   Worked Example

   A cone has a base diameter of 12 m and a perpendicular height of 8 m. Calculate its volume in terms of $\pi$.

   1. Determine the radius: $r = \frac{12}{2} = 6\text{ m}$.
   2. Substitute into the volume formula: $V = \frac{1}{3}\pi (6)^2 (8)$.
   3. Evaluate: $V = \frac{1}{3}\pi (36)(8) = \frac{288\pi}{3} = 96\pi\text{ m}^3$.

   Understanding Spherical Volume

   A sphere is the set of all points in three-dimensional space that are equidistant from a fixed central point. The distance from the center to any point on the surface is the radius $r$. Unlike cylinders and cones, a sphere has no base or uniform height.

   The formula for the volume $V$ of a sphere is:

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

   Hemispheres

   A hemisphere is exactly one-half of a sphere created by a plane passing through the center. Its volume is half the volume of a complete sphere:

   $$V_{\text{hemisphere}} = \frac{1}{2} \left(\frac{4}{3}\pi r^3\right) = \frac{2}{3}\pi r^3$$

   Worked Example

   A sphere has a radius of 3 cm. Find its volume.

   • Substitute $r = 3$ into the formula: $V = \frac{4}{3}\pi (3)^3$
   • Simplify the power: $3^3 = 27$
   • Multiply: $V = \frac{4}{3}\pi (27) = 36\pi\text{ cm}^3 \approx 113.1\text{ cm}^3$

   Key Tip: Ensure that the radius is cubed ($r^3$), not squared ($r^2$), when working with spheres.

   Geometric Relationships and Summary

   Understanding the proportional relationships among these three fundamental curved solids reinforces conceptual mastery and aids in problem solving:

   • Cone vs. Cylinder: If both have the same radius $r$ and height $h$, their volume ratio is $1 : 3$.
   • Sphere inside a Cylinder: Consider a sphere of radius $r$ enclosed tightly inside a cylinder. The cylinder must have radius $r$ and height $h = 2r$. Its volume is $V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3$. Comparing this to the sphere's volume $V_{\text{sphere}} = \frac{4}{3}\pi r^3$, we find that the sphere occupies exactly $\frac{2}{3}$ of the cylinder's volume.

   Common Misconceptions to Avoid

   1. Radius vs. Diameter: Forgetting to halve the diameter before calculating $r^2$ or $r^3$.
   2. Power of the Radius: Squaring instead of cubing the radius in the sphere formula, or cubing instead of squaring in the cylinder/cone formulas.
   3. Units: Volume is always expressed in cubic units (e.g., $\text{cm}^3$, $\text{m}^3$), whereas surface area is in square units.