Volume of Solids
Measurements · Volume of Solids
Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 3.0 Measurements | Sub-Strand 3.2 Volume of Solids (8 Lessons)
Lesson objectives
By the end of this sub-strand, you should be able to:
- Work out the volume of prisms, cylinders, pyramids, cones and spheres.
- Recognise which solids are uniform-cross-section solids and which taper to a point.
- Convert confidently between cubic units and litres.
- Apply volume calculations to practical problems involving tanks, containers and materials.
Volume of Solids
Volume measures the space inside a solid. A water tank, a grain store and a tin of cooking fat all hold a measured amount. Volume is how we work that amount out.
Every solid in this topic uses one of only three ideas. Learn the three and you can handle any of them.
a) Volume of a prism
A prism has the same cross-section from one end to the other. A brick, a tent and a length of timber are all prisms.
There is one rule for every prism, whatever shape its end is.
If the end is a triangle instead of a rectangle, only the first step changes. Work out the triangle's area, then multiply by the length as before.
Watch the units. Two lengths multiplied give cm², and three give cm³. If your answer for a volume is in cm², you have missed a step.
b) Volume of a pyramid
A pyramid has one base and faces that meet at a point.
A pyramid holds exactly one third of the prism that would fit around it. Same base, same height, one third of the space.
c) Volume of a cone
A cone is a pyramid whose base is a circle, so the same one third applies.
Use the vertical height here, measured straight up from the centre of the base. The slant height belongs to surface area, not volume.
d) Volume of a frustum
Cut the top off a cone and the piece left behind is a frustum. A bucket and a lamp shade are both frustums.
You cannot measure it directly, so work by subtraction. Find the volume of the whole cone before it was cut. Find the volume of the small cone that was removed. Subtract.
e) Volume of a sphere
A sphere is round in every direction, like a ball.
Its volume is four thirds of pi times the radius cubed. There is no simpler solid to build it from, so this one has to be learnt.
Note that the radius is cubed, not squared. Doubling the radius makes the volume eight times larger, not twice.
f) Where this is used
A water engineer sizes a tank by volume before ordering it. A jua kali fundi making a bucket works with a frustum. A trader buying maize by the gorogoro is measuring volume every time.
Words to know
- Volume -- the amount of space occupied by a solid, expressed in cubic units.
- Prism -- a solid with the same cross-section throughout its length.
- Cross-section -- the flat shape revealed by cutting a solid straight across.
- Perpendicular height -- the vertical distance from base to apex, measured at right angles to the base.
- Displacement -- the method of finding volume by measuring the water a submerged object pushes aside.
:::checkpoint Check yourself
- A rectangular prism measures 10 cm by 6 cm by 3 cm. What is its volume?
- A cone has radius 7 cm and vertical height 9 cm. Find its volume.
- Which height do you use when finding the volume of a cone?
- How do you find the volume of a frustum? :::
Bridge to practice
The exercises begin with direct prism and cylinder calculations, move through cones, pyramids and spheres, and finish with composite solids and unit conversion in a capacity context. Before substituting, write down which family the solid belongs to and which measurements you have actually been given; several questions below are built to catch a diameter used as a radius.