Mensuration — 2D Figures

Areas and perimeters of the plane figures an exam actually asks for — rectangle, square, parallelogram, rhombus, trapezium, triangle (base–height, Heron's formula, equilateral, right-angled) and circle; sector, segment and ring; paths inside and around a field; effect on area when a dimension changes by a percentage; converting between square metres, hectares and acres; cost problems on fencing, flooring and turfing — with fully worked examples.

Mensuration — 2D Figures

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📑 Contents (8 sections)

Last reviewed 22 Sept 2026 · 5 min read

Quadrilaterals

FormulaAreas and perimeters

Figure	Area	Perimeter
Rectangle	$l \times b$	$2 (l + b)$ , diagonal $l^{2} + b^{2}$
Square	$a^{2} = \frac{d ^{2}}{2}$	$4 a$ , diagonal $a 2$
Parallelogram	base $\times$ height	$2 (a + b)$
Rhombus	$\frac{1}{2} d_{1} d_{2}$	$4 a$ , with $a = \frac{1}{2} d_{1}^{2} + d_{2}^{2}$
Trapezium	$\frac{1}{2} (a + b) h$	sum of the four sides

Triangles

FormulaFour ways to an area

Base and height: $\frac{1}{2} bh$
Three sides (Heron): $s = \frac{a + b + c}{2}$ , area $= s (s - a) (s - b) (s - c)$
Equilateral of side $a$ : $\frac{3}{4} a^{2}$ , height $\frac{3}{2} a$
Right-angled: $\frac{1}{2} \times$ product of the two legs

Circle, sector, segment and ring

FormulaCircle formulas

For radius $r$ and an angle $θ$ in degrees:

Area = π r^{2}, Circumference = 2 π r

Arc = \frac{θ}{360} \times 2 π r, Sector area = \frac{θ}{360} \times π r^{2} = \frac{arc \times r}{2}

Minor segment = sector - triangle, Ring (annulus) = π (R^{2} - r^{2})

Use $π = \frac{22}{7}$ when the radius is a multiple of 7, and $3.14$ otherwise.

Paths and borders

A path of width $w$ outside a rectangle $l \times b$ : outer size $(l + 2 w) \times (b + 2 w)$ ; path area $=$ outer $-$ inner.
A path inside: inner size $(l - 2 w) \times (b - 2 w)$ .
Two crossing paths through the middle of a field: area $= l w + b w - w^{2}$ — the overlap is counted once.

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