Area of Quadrilaterals and Other Polygons

Syllabus tag: KCSE | Mathematics | Form 2 | Topic 11 Area of Quadrilaterals and Other Polygons

Lesson objectives

By the end of this topic, you should be able to:

• Find the area of quadrilaterals using appropriate formulae.
• Find the area of regular polygons by dividing them into triangles.
• Find the area of irregular polygons by decomposition.
• Apply these methods to real situations.

Area of Quadrilaterals and Other Polygons

Every polygon's area can be found by breaking it into shapes you already know.

a) The quadrilateral formulas

For the parallelogram, the height is the perpendicular distance between the parallel sides, never the slanting side.

For the rhombus and the kite, the diagonals cross at right angles. That is why half their product gives the area.

For the trapezium, add the parallel sides first, then halve, then multiply by the perpendicular height.

A square is a special rectangle, so s². A rhombus may also be treated as a parallelogram, b × h, when a height is given.

b) Choosing the formula

Look at what the question gives you.

Diagonals given points to the rhombus or kite formula. A perpendicular height points to base times height. Two parallel sides point to the trapezium.

c) Regular polygons

A regular polygon splits into n identical triangles, one for each side, meeting at the centre.

The apothem is the perpendicular distance from the centre to a side. It is the height of each triangle.

So area = n × ½ × side × apothem, which simplifies to ½ × perimeter × apothem.

d) Finding the apothem

Each central triangle has an apex angle of 360/n.

Split it in half. That gives a right-angled triangle with angle 180/n at the centre, and half the side opposite.

So the apothem is (half the side) ÷ tan(180/n).

For a hexagon of side 6: half the side is 3, and 180/6 is 30°. So the apothem is 3 ÷ tan 30° = 5.196.

e) Irregular polygons

Split the shape into triangles, rectangles and trapeziums.

Work out each part, then add.

Where a piece is cut out, subtract instead.

Sketch the split and label each part before calculating. That prevents double counting.

There is usually more than one valid way to split a shape. Any correct split gives the same total.

f) A useful alternative

Any polygon can be split into triangles by joining one vertex to all the others.

An n-sided polygon gives n − 2 triangles. That is also why the interior angles sum to 180(n − 2).

If the sides are known but no heights, use Heron's formula on each triangle.

g) Where this is used

Land measurement, where plots are rarely regular. Costing flooring and roofing. Estimating material for irregular panels.

Words to know

• Regular polygon -- one with all sides and all angles equal.
• Apothem -- the perpendicular distance from the centre of a polygon to a side.
• Decomposition -- splitting a figure into simpler shapes whose areas are known.
• Offset -- a perpendicular distance measured from a base line to a boundary.
• Central angle -- the angle at the centre subtended by one side, equal to 360° ÷ n.

:::checkpoint Check yourself

1. Find the area of a trapezium with parallel sides 9 cm and 15 cm and height 6 cm.
2. Find the area of a rhombus with diagonals 10 cm and 8 cm.
3. Into how many triangles does a heptagon split from one vertex?
4. What is the apothem of a regular polygon? :::

Bridge to practice

The exercises begin with the quadrilateral formulae and their derivations, move through regular polygons by both methods, and finish with decomposition and land measurement. For any unfamiliar figure, look first for a split that uses the measurements you have been given.