Area
Measurements · Area
Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 3.0 Measurements | Sub-Strand 3.1 Area (8 Lessons)
Lesson objectives
By the end of this sub-strand, you should be able to:
- Work out the area of a circle, a sector and a segment.
- Calculate the surface area of cylinders, cones, spheres and other common solids.
- Break irregular shapes into familiar parts in order to find their total area.
- Apply area calculations to practical problems involving land, materials and cost.
Area
Area measures the surface of a flat shape. Surface area measures the outside of a solid one. A painter buying paint and a tinsmith cutting sheet metal both need these numbers.
Every shape in this topic breaks down into shapes you already know. That is the whole method.
a) Area of a pentagon and a hexagon
A regular polygon has equal sides and equal angles. Draw lines from the centre to every corner and it falls apart into identical triangles.
A hexagon gives six triangles. A pentagon gives five. Find the area of one triangle, then multiply.
The height used here runs from the centre to the middle of a side. It is not the slanted edge. Using the edge by mistake gives an answer that is too big.
b) The area of a sector
A sector is the slice of a circle between two radii. Think of a slice cut from a chapati.
A sector is a fraction of the whole circle. A 90° sector is a quarter. A 100° sector is 100 out of 360.
c) The area of a segment
A segment is the piece cut off by a straight chord.
There is one method, and it is subtraction. Work out the sector. Work out the triangle formed by the two radii and the chord. Take the triangle away from the sector.
The segment is always smaller than its sector. If your answer is bigger, you have added instead of subtracted.
d) Surface area of prisms
A prism has the same shape all the way through, like a tent or a length of timber.
To find its surface area, imagine cutting it open and laying it flat. That flat pattern is called a net.
Every prism gives the same pattern: two identical ends, plus one long rectangle wrapped round. The rectangle's width is the perimeter of the end.
e) Surface area of pyramids
A pyramid has one base and triangular faces meeting at a point.
Add the base to all the triangles. A square-based pyramid has four identical triangles, so work out one and multiply by four.
Use the slant height for these triangles. That is the distance up the sloping face, not the vertical height of the pyramid. Mixing them up is the usual error.
f) Cones and spheres
A cone is like a pyramid with a circular base. A sphere has no base at all.
The sphere formula is worth memorising. There is no way to build it from simpler shapes at this level.
g) Where this is used
A roofer works out surface area to order iron sheets. A jua kali metalworker cuts nets before bending a water tank. A farmer painting a cylindrical silo needs the curved surface to know how much paint to buy.
Words to know
- Area -- the measure of the flat surface enclosed by a shape, expressed in square units.
- Sector -- the region of a circle bounded by two radii and the arc between them.
- Segment -- the region of a circle cut off by a chord.
- Slant height -- the distance measured along the sloping surface of a cone from the base edge to the apex.
- Composite shape -- a figure made by combining or removing simpler shapes.
:::checkpoint Check yourself
- A regular pentagon has sides of 10 cm and a height of 6.9 cm from centre to side. What is its area?
- Find the area of a 60° sector of a circle of radius 12 cm.
- How do you find the area of a segment?
- Which height do you use for the triangular faces of a pyramid? :::
Bridge to practice
The exercises begin with direct circle and sector calculations, move through cylinder and cone surfaces, and finish with composite figures and unit conversion in a costing context. For every question, note first which measurement you have been handed and whether the solid described is open or closed, since those two checks determine almost everything that follows.