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Volume of cones, pyramids and spheres

Optional extension for this course. It is not required for core topic progress.

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A right cone and a cylinder have the same circular base and perpendicular height. Which statement is correct?

Year 10, Chapter 1, 4 difficulty levels, Calculator allowedBuild a worksheet

Advanced solid volumes

Distinguish perpendicular height from slant height, radius from diameter, and internal capacity from solid material. Add only non-overlapping component volumes

\[\begin{gathered}V_{\text{pyramid/cone}}=Bh/3\\V_{\text{sphere}}=4\pi r^3/3\\V_{\text{hemisphere}}=2\pi r^3/3\end{gathered}\]
  1. Same base and height The one-third comparison is with the prism or cylinder having exactly the same base area and perpendicular height
  2. Length scale factor k For similar solids, volume is multiplied by k³
  3. Shared boundaries A contact face is not a volume. Genuine overlapping volumes are counted once
  4. Units and precision Convert all lengths to a common unit first. Use cubic units and round once at the end

Worked example

A solid cylinder with radius 3 cm and height 5 cm is capped by one hemisphere with the same radius. Find the combined volume to 1 decimal place

\[V=V_c+V_h\]
  1. \(\displaystyle V_c=\pi(3)^2(5)=45\pi\text{ cm}^3\)Cylinder volume
  2. \(\displaystyle V_h=2\pi(3)^3/3=18\pi\text{ cm}^3\)One hemisphere, not a whole sphere
  3. \(\displaystyle V=(45+18)\pi=63\pi\text{ cm}^3\)The interiors do not overlap
  4. \(\displaystyle V\approx197.9\text{ cm}^3\)Round the total once

Common mistakes

Using slant height for volume
✕\(\displaystyle V=\pi r^2s/3\)✓\(\displaystyle V=\pi r^2h/3\)
Here s is slant height and h is perpendicular height. They are different lengths
Halving the radius of a hemisphere
✕\(\displaystyle V=4\pi(r/2)^3/3\)✓\(\displaystyle V=(4\pi r^3/3)/2\)
Cutting the sphere in half preserves its radius and halves its volume
Subtracting contact area
✕\(\displaystyle V=V_1+V_2-\pi r^2\)✓\(\displaystyle V=V_1+V_2\)
For parts joined only at a face, adding cubic volumes is correct. Square-unit contact area cannot be subtracted from volume

Difficulty levels

introBuild a pyramid volume from its base area
standardCalculate sphere volumes from radius or diameter
challengeUse cones, hemispheres, stated base areas and composite volumes
examDistinguish correct sphere and hemisphere calculations from fraction and radius errors

Questions about this topic

Does the one-third rule need a circular base?+

No. A pyramid uses one third of the matching prism volume for the same base area and perpendicular height; a cone uses the corresponding cylinder

Is a rounded ratio of 0.333 a proof?+

No. It is numerical evidence consistent with one third. The formulas give the exact ratio after the common base area and height cancel

Does a hemisphere have half the radius?+

No. It has the same radius as the sphere from which it is cut, and half that sphere’s volume

Can I halve an already rounded sphere volume?+

Keep the exact sphere expression and halve that. Rounding an intermediate can change the final rounded result

When does a composite volume need subtraction?+

Subtract space that is removed or a three-dimensional overlap counted twice. Merely touching along a face creates no overlap of volume

Are these geometric volumes automatically usable capacities?+

Only if the dimensions describe internal usable space and the ideal-shape assumptions are justified. Wall thickness, fittings and filling limits may reduce actual capacity

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 1: Measurement

In this chapter

Convert measurement unitsPerimeter of standard shapesFind an unknown side from a perimeterCircumference of a circlePerimeter of composite shapesArea of standard shapesArea of a circleArea of a sectorArea of composite shapes1F Prism surface area: Surface area of solidsSurface area of a triangular prismExposed surface area of joined cuboids1G Cylinder surface area: Surface area of solidsSurface area of a cylinder portionSurface area of a cylinder joined to a cuboidVolume of prisms and cylindersCapacity of a containerVolume of a cylinder portionJoined and hollow prism-cylinder volumesModel rainwater collection and storageSmallest unit and limits of accuracyMeasurement bounds in calculationsVolume of cones, pyramids and spheres (optional extension)1J Level 10A extension: pyramids, cones and spheres: Surface area of solids (optional extension)Surface areas of advanced solids and composites (optional extension)