Learn: how much fits inside?
Imagine a cone-shaped mould with an inside radius of 3 cm and a perpendicular height of 4 cm. Volume measures the space inside it. Paper needed to cover it is a different question: that is surface area.
The radius r runs from the centre of the circular base to its edge. The diameter d = 2r runs across the entire base. The height h is the perpendicular distance from the tip to the plane of the base; it is not the distance along the sloping side.
For a circular cone, V = (1/3)πr²h. Here V is volume, π (pi) is approximately 3.14159, and r² means r × r. Use the same length unit for r and h; volume then has cubic units such as cm³.
Why compare it with a cylinder?
A cylinder with the same base radius and perpendicular height has volume πr²h. The cone occupies exactly one third of that volume. Its cross sections get smaller toward the tip in both horizontal directions. It is not half the cylinder merely because a side-view triangle has half a rectangle’s area.
The volume rule also applies to an oblique circular cone when h is the perpendicular height. The sketch here shows a right circular cone, with the tip directly above the base centre.
Worked example: r = 3 cm, h = 4 cm
- Find the base area: π × 3² = 9π cm².
- Multiply by height: 9π × 4 = 36π cm³ for the matching cylinder.
- Take one third: V = 12π cm³ ≈ 37.70 cm³.
This ideal cone holds about 37.70 mL because 1 cm³ = 1 mL. A real container’s wall thickness and fill level matter; use its inside dimensions.
Worked example: diameter 10 cm, height 15 cm
First r = 10 ÷ 2 = 5 cm. Then V = (1/3) × π × 5² × 15 = 125π cm³ ≈ 392.70 cm³. Substituting 10 as the radius would give four times the correct volume.
Explore: change one dimension
The diagram is schematic, not to scale. Quantities below it update when you move the sliders. Predict the change first, then compare the result.
Try radius 3 cm, then 6 cm. For volume, keep the height fixed where available. Explain why the change differs from simply doubling the result. The shape drawing stays fixed so you can focus on the labelled dimensions and numerical comparison.
Practice: explain before checking
Work each question on paper. Open the explanation to compare your setup, calculation and units.
1. A cone has r = 2 in and h = 6 in. Find its volume.
V = (1/3)π(2²)(6) = 8π in³ ≈ 25.13 in³. Square the radius before multiplying by height.
2. Double the radius but keep the height. What changes?
The volume becomes four times as large because (2r)² = 4r². Doubling only the height would double the volume.
3. The sloping side is 5 cm, the radius is 3 cm, and the cone is right. Is 5 the height?
No. The radius, perpendicular height and slant height form a right triangle. h² = 5² − 3² = 16, so h = 4 cm. Volume is 12π cm³.
Review
Use the perpendicular height, halve a given diameter, divide the matching cylinder volume by three, and report cubic units.
Recall without looking: What does each symbol measure? Which units belong in the answer? Which surfaces or spaces are included?