Volume of Solids
Measurement · Volume of Solids
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 14 Volume of Solids
Lesson objectives
By the end of this topic, you should be able to:
- Find the volume of prisms and pyramids.
- Find the volume of a cone and a frustum.
- Find the volume of a sphere and hemisphere.
- Solve problems involving composite solids and displacement.
Volume of Solids
Volume is the space a solid occupies, measured in cubic units.
a) Prisms
A prism has the same cross-section along its whole length.
Volume = area of cross-section × length.
A cuboid is a prism with a rectangular cross-section, so V = lwh.
A cylinder is a prism with a circular cross-section, so V = πr²h.
A triangular prism uses the triangle's area times the length.
That single rule covers all prisms. Learn it once rather than three formulas.
b) Pyramids and cones
A pyramid has a base and triangular faces meeting at an apex.
Volume = ⅓ × base area × vertical height.
A cone is a pyramid with a circular base, so V = ⅓πr²h.
The third is not arbitrary. Three pyramids of the same base and height fill the matching prism exactly.
Use the vertical height here, not the slant height. Slant height belongs to surface area; vertical height belongs to volume.
c) Spheres and hemispheres
A sphere has volume ⁴⁄₃πr³.
A hemisphere is half of that, so ⅔πr³.
Note the cube. Volume depends on r³, so doubling the radius multiplies volume by eight.
d) Frustums
A frustum is a cone or pyramid with the top cut off parallel to the base.
Its volume is the whole cone's volume minus the removed top cone's.
Find the removed cone's dimensions from similar triangles. The radii and heights are in the same ratio.
e) Composite solids
Split the solid into parts you recognise, work out each, then add.
Where a piece has been hollowed out, subtract instead.
A common example is a cylinder with a hemisphere on top: add ⅓ the two volumes as appropriate.
Sketch the split and label each part before calculating.
f) Displacement
An object placed in water raises the level. The volume of water displaced equals the volume of the object.
This gives the volume of an irregular solid such as a stone.
Measure the water level before and after. The difference in volume is the object's volume.
If the container is a cylinder, that difference is πr² × rise in level.
g) The formulas together
h) Where this is used
Sizing tanks and silos. Costing concrete. Finding the volume of an irregular object. Estimating how much a container holds.
Words to know
- Prism -- a solid with the same cross-section throughout its length.
- Vertical height -- the perpendicular distance from apex to base.
- Frustum -- a cone or pyramid with its top removed parallel to the base.
- Displacement -- the volume of liquid pushed aside by a submerged solid.
- Conservation of volume -- the principle that recasting changes shape but not volume.
:::checkpoint Check yourself
- Find the volume of a cylinder with radius 7 cm and height 10 cm, taking π as 22/7.
- Find the volume of a cone with the same radius and height.
- Find the volume of a sphere of radius 3 cm, taking π as 3.142.
- Why does a pyramid have one third the volume of the matching prism? :::
Bridge to practice
The exercises begin with prisms and the one-third rule, move through cones, spheres and frustums, and finish with composite solids, displacement and recasting. For every cone or pyramid, write down whether the height given is vertical or slant before starting.