Methods and worked examples: Area

2.1 The triangle formula holds for every triangle

For a right or acute triangle, enclose it in a rectangle with the same base and height and the formula follows at once. For an obtuse triangle, the perpendicular from the top vertex falls outside the base. In that case the triangle is the difference between two right triangles that share the same height, and the same formula comes out after simplifying. So it is safe to use area = ½ × base × height for every triangle, as long as the height is measured perpendicular to the chosen base.

Any of the three sides can be taken as the base; each choice comes with its own height. A triangle has one area, but three different base-height pairs, and the product base × height is the same for all three pairs. This is why a question that gives two sides and one altitude usually lets you find the other altitude: write the area twice and equate the expressions.

2.2 Finding a height from the area

The chapter's example gives a triangle with AX = 5 units, BC = 3 units and AC = 4 units, where AX is perpendicular to BC and BY is perpendicular to AC. The area from the first pair is ½ × 3 × 5 = 7.5 square units. Using AC as the base, the same area is ½ × 4 × BY = 2 BY. Equating the two gives 2 BY = 7.5, so BY = 3.75 units.

The method is always the same: compute the area from a known base-height pair, then write the area a second time with the unknown height. This is a common exam pattern.

2.3 Triangles with equal areas

A line from a vertex to the midpoint of the opposite side divides a triangle into two triangles of equal area, because the two halves have equal bases and share the same height. Drawn twice, this explains why the diagonals of a rectangle cut it into four triangles of equal area: the diagonals bisect each other, so each small triangle has a base that is half a diagonal and the same height.

Triangles that stand on the same base between two parallel lines also have equal areas, because their heights are the same. Their perimeters can differ a lot: the triangle that leans furthest has the longest perimeter even though its area does not change. The shortest perimeter is attained by the triangle whose apex lies on the perpendicular bisector of the base; the reflection argument with the mirror line proves it.

The same idea lets us find areas of polygons. Joining one vertex of a polygon to the other vertices splits it into triangles, and adding their areas gives the area of the polygon. A quadrilateral needs one diagonal, a pentagon needs two, and in general a polygon with n sides splits into n − 2 triangles.

2.4 The parallelogram, rhombus and trapezium

A parallelogram can be cut along a perpendicular from a vertex and rearranged into a rectangle. The rectangle has the same base and the same height, so the parallelogram's area is base × height. A rhombus is a special parallelogram, and cutting it along its diagonals shows another formula: its area is half the product of the two diagonals.

A trapezium comes apart into a rectangle and two triangles. Writing the three areas and combining them with letters gives area = ½ × height × (sum of the parallel sides). The formula can also be obtained by joining two copies of the trapezium to form a parallelogram. All four formulas are collected in the revision page.

Areas in real life use the same rules with units converted first. One inch is 2.54 cm, so one square inch is 2.54 × 2.54 = 6.4516 cm². One foot is 12 inches and one acre is 43,560 square feet. A worked example: an A4 sheet is about 21 cm by 29.7 cm, so its area is 21 × 29.7 = 623.7 cm². A tabletop 90 cm by 60 cm has area 5,400 cm², which is about 8.7 A4 sheets. The comparison uses area, not length: the tabletop is not much longer than the sheet, yet it holds several sheets because both dimensions are larger. This kind of estimate is useful in daily life, where lengths are easy to measure but the number of units that fit inside is not visible. A rough estimate is often enough to decide whether a measured answer is plausible, even before the exact calculation is finished. A classroom 8 m by 6 m has area 48 m², which is 480,000 cm²; the unit conversion multiplies by 10,000 because a square metre holds 100 by 100 square centimetres. Convert lengths before multiplying, because a length conversion becomes a squared factor for area. A useful habit is to estimate the area first, then check that the calculated value has the right order of magnitude.