6. Volume of a Cone

Session 6 Learning Plan

Learning objective: Calculate cone volume and explain the one-third relationship.

Key vocabulary: cone, apex, perpendicular height, matching cylinder, one-third

Success criterion: Apply \(V=\tfrac13\pi r^2h\) and verify the result against a matching cylinder.

Learn and Practice

A cone has a circular base and narrows to a point. It holds exactly one-third as much as a cylinder with the same base and height:
\[ V = \tfrac{1}{3} \pi r^2 h. \]

Worked example: a cone with radius 3 cm and height 4 cm has volume
\[ V = \tfrac{1}{3} \times 3.14 \times 3^2 \times 4 = \tfrac{1}{3} \times 3.14 \times 9 \times 4 = \tfrac{1}{3}(113.04) = 37.68 \text{ cm}^3. \]

A good shortcut: compute the cylinder volume first (\(\pi r^2 h\)), then divide by 3.

Case study - an ice cream cone or a paper cup: its capacity uses the cone formula, which is why a cone holds less than a can of the same size.

Common mistake: forgetting the \(\tfrac{1}{3}\). Without it you get the cylinder's volume, which is three times too big.

Why the Cone Uses One-Third

A cone and a cylinder with the same radius and perpendicular height do not hold the same amount. Three matching cone volumes fill the cylinder, so
\[ V_{\text{cone}}=\tfrac13V_{\text{cylinder}}=\tfrac13\pi r^2h. \]
This comparison is the best protection against forgetting the factor ( frac13).

For diameter \(12\text{ m}\) and height \(9\text{ m}\), use \(r=6\text{ m}\):
\[ V=\tfrac13\pi(6)^2(9) =\tfrac13\pi(36)(9) =108\pi\text{ m}^3. \]
An answer of \(324\pi\) is the matching cylinder volume and reveals the missing division by \(3\).

Height Is Not Slant Height

The volume formula uses the perpendicular height from the center of the circular base to the tip. A slant height runs along the side and belongs in some surface-area formulas, not cone volume. If both are labeled, mark the right-angle relationship to identify the perpendicular height.

Scaling and Estimation

If radius doubles while height stays fixed, cone volume becomes four times as large because \(r^2\) changes by \(2^2\). If both radius and height double, volume changes by \(2^2\times2=8\). Before calculating a decimal, estimate from the matching cylinder: the cone answer must be positive and exactly one-third of that cylinder's volume. This relationship catches radius, exponent, and fraction errors in one check.

GED® Coaching: A Reliable Five-Pass Method

Use the same disciplined process on every three-dimensional measurement problem.

1. Name the target. Write volume or surface area before calculating. Words such as hold, fill, capacity, and inside usually signal volume. Words such as cover, wrap, paint, label, and outside usually signal surface area.
2. Identify the solid and the given measurements. Mark length, width, height, radius, diameter, base area, or slant height. Never assume a drawing is to scale. If a diameter is given but the formula uses radius, write \(r=d/2\) before substituting.
3. Write the formula in symbols. Use the GED® formula sheet, but choose the formula from the meaning of the problem. Then substitute values with units. This makes a missing square, cube, or one-third factor much easier to spot.
4. Calculate in visible stages. Evaluate exponents first, then multiplication and division. Keep \(\pi\) in the calculator until the final step unless the question directs you to use \(3.14\). Round only once, at the end.
5. Verify meaning, size, and unit. Volume must use cubic units; surface area must use square units. Estimate with friendly numbers and ask whether the answer fits the situation. A cone matching a cylinder must have one-third of its volume, and a closed container must include every exposed face.

When an answer is wrong, classify the cause rather than merely recording the score: target error, solid-identification error, formula error, radius/diameter error, exponent error, unit error, calculation error, or rounding error. Rewrite the solution from the first incorrect decision. This correction routine turns each missed item into a reusable GED® strategy.

Session Practice Routine

1. State whether the target is interior volume or exterior surface area.
2. Name the solid and mark only the required measurements.
3. Convert dimensions to compatible units before using a formula.
4. Write the formula, substitute visibly, and evaluate in stages.
5. Attach square or cubic units and verify the size by estimation.
6. Complete the interactive self-check, then explain any correction aloud.

Exit Reflection

Without looking back, sketch the session's solid or net, write the key formula
from meaning, name one likely error, and describe the check that would catch it.