Cone volume: vertical height is not slant height

Draw the cross-section before choosing a height

For a right circular cone, the tip is directly above the centre of its base. The perpendicular height runs from that tip to the centre. The slant height runs along the surface to the rim. Radius, perpendicular height and slant height form a right triangle in a central cross-section.

The volume formula uses perpendicular height because volume measures how the circular cross-sections stack above the base. Substituting the longer slant height overestimates the volume. Draw h inside the cone and l along its edge; the letters identify different measurements.

Understand the one-third factor

A cone’s volume is one third of a cylinder with the same circular base and perpendicular height. For radius 3 cm and height 8 cm, the cylinder volume is 72π cm³ and the cone volume is 24π cm³. The comparison requires both dimensions to match; any randomly chosen cone and cylinder will not have this relationship.

For an ideal hollow container, volume gives capacity. Real wall thickness and how high the container is filled can change the usable capacity. State that you are modelling the inside shape when using a cone formula for a cup or funnel.

Find a missing dimension before the volume

If radius is 5 cm and slant height is 13 cm, h²=13²−5²=144, so h=12 cm. Choose the positive root because a physical length is positive. Now V=π×25×12/3=100π cm³.

If volume is 48π cm³ and radius is 4 cm, rearrange 48π=π×16×h/3. Cancel π and multiply by 3: 144=16h, so h=9 cm. Substitution checks the result: 16×9/3=48.

Keep volume separate from the material needed

Volume answers how much space is inside. Curved surface area answers how much material covers the sloping surface. For a closed cone, add the base area to the curved surface area. A funnel open at its base is a different surface question from a closed solid cone.

If all lengths scale by two, volume scales by 2³=8. If only the radius doubles and height stays fixed, volume scales by four. If only height doubles, volume doubles. State which dimensions change before applying a scale rule.

Convert units before substituting

A radius of 20 mm and height of 6 cm cannot be inserted as 20 and 6 without conversion. Use radius 2 cm, giving V=π×4×6/3=8π cm³. One cubic centimetre equals 1,000 cubic millimetres; the factor is cubed because volume has three dimensions.

A diagram need not be to scale, so read labels rather than estimating lengths from its appearance. A claimed slant height shorter than the radius or perpendicular height cannot describe a right cone and should be checked before calculating.

Worked examples: follow each decision

13 questions with explained answers

Attempt a short group of questions before opening the solutions. Use a drawing or written method where helpful. Explain the answer, then try a similar question with a detail changed.

1. r = 2 cm, h = 9 cm: exact volume?

   Show answer and reasoning

   12π cm³

   (1/3) × π × 4 × 9.

2. r = 6 cm, h = 8 cm: slant height?

   Show answer and reasoning

   10 cm

   √(36 + 64) = 10.

3. Same r and h: cone versus cylinder volume?

   Show answer and reasoning

   Cone is one third

   Both use the same base area and height.

4. Double height only: effect on cone volume?

   Show answer and reasoning

   Volume doubles

   The formula is directly proportional to h.

5. Double radius only: effect on volume?

   Show answer and reasoning

   Volume becomes four times as large

   The radius is squared.

6. r=4 cm, h=9 cm: exact volume?

   Show answer and reasoning

   48π cm³

   π×16×9/3.

7. r=8 cm, l=17 cm: height?

   Show answer and reasoning

   15 cm

   √(289−64)=15.

8. V=75π cm³ and h=9 cm: radius?

   Show answer and reasoning

   5 cm

   r²=3×75/9=25.

9. Diameter 6 cm, height 10 cm: volume?

   Show answer and reasoning

   30π cm³

   Use radius 3, not 6.

10. All dimensions triple: volume factor?

    Show answer and reasoning

    27

    3³=27.

11. r=20 mm, h=6 cm: volume in cm³?

    Show answer and reasoning

    8π cm³

    20 mm=2 cm.

12. Could r=6 cm and l=5 cm describe a right cone?

    Show answer and reasoning

    No

    The slant height is the hypotenuse and must exceed the radius.

13. Which uses slant height: volume or curved surface area?

    Show answer and reasoning

    Curved surface area

    πrl uses l; volume uses perpendicular h.

Check what you can now explain

• Is your height perpendicular or sloping?
• Did you halve a given diameter?
• Are you measuring space, capacity or covering material?

Keep these distinctions clear

• Volume uses cubic units, not square units.
• The right-triangle relation requires a right circular cone, with the tip above the base centre.

Save a useful copy

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Keep the ideas connected

• Grade 8 Pythagoras’ Theorem and the Right-Triangle Test
• Grade 8 Cylinder Surface Area and Volume
• Choose a Maths practice game by skill

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