Mensuration (2D)

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high importance~2 Q in Tier 126 formulas⚡ 15 shortcuts5 subtopics

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Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.

Areas of triangles

Triangle area

K=\frac{1}{2}bh

b = any side, h = perpendicular height on it.

Equilateral triangle

K=\frac{\sqrt3}{4}a^2,\quad h=\frac{\sqrt3}{2}a

a = side; height is root-three over two of the side.

Heron's formula

K=\sqrt{s(s-a)(s-b)(s-c)},\quad s=\frac{a+b+c}{2}

s = half the perimeter.

Altitude to hypotenuse

h=\frac{ab}{c}=\frac{2K}{c}

a, b legs, c hypotenuse; from equating two areas.

Inradius / circumradius

r=\frac{K}{s},\quad R=\frac{abc}{4K}

K = area, s = half-perimeter.

Median split

\text{median}\Rightarrow\text{two equal areas}

Each median halves the area of a triangle.

Areas of quadrilaterals

Rectangle

K=lb,\quad P=2(l+b),\quad d=\sqrt{l^2+b^2}

Three linked facts; any two fix the third.

Square

K=a^2,\quad d=a\sqrt2

Diagonal is root-two times the side.

Parallelogram

K=bh=ab\sin\theta

theta = angle between the two given sides.

Rhombus

K=\frac{1}{2}d_1d_2

Diagonals cross at right angles and halve each other.

Trapezium

K=\frac{1}{2}(a+b)h

a, b = the two parallel sides.

Any quadrilateral

K=\frac{1}{2}d(h_1+h_2)

d = a diagonal; h1, h2 = perpendiculars onto it.

Circles, sectors and rings

Circle

K=\pi r^2,\quad C=2\pi r

Take pi as 22/7 unless stated.

Arc and sector

\text{arc}=\frac{\theta}{360}2\pi r,\quad \text{sector}=\frac{\theta}{360}\pi r^2

Same fraction of circumference and area.

Ring

K=\pi(R^2-r^2)

R = outer radius, r = inner radius.

Wheel revolutions

N=\frac{D}{C}=\frac{\text{distance}}{2\pi r}

Count turns by dividing distance by one circumference.

Quadrant / semicircle

\text{quad}=\frac{\pi r^2}{4},\quad \text{semi}=\frac{\pi r^2}{2}

Quarter and half of the circle area.

Regular polygons and inscribed figures

Regular hexagon

K=\frac{3\sqrt3}{2}a^2

Six equilateral triangles of side a.

Polygon from apothem

K=\frac{1}{2}\times P\times a

P = perimeter, a = apothem (centre to a side).

Exterior angle

\text{ext}=\frac{360^\circ}{n}

Equal turns around the boundary.

Interior angle

\text{int}=180^\circ-\text{ext},\quad \text{sum}=(n-2)180^\circ

One angle plus the total for all n.

Diagonals

d=\frac{n(n-3)}{2}

n sides give this many diagonals.

Percentage change, similarity and re-bent shapes

Similar figures

\frac{a_1}{a_2}=k\Rightarrow\frac{K_1}{K_2}=k^2

Lengths take k once; areas take k squared.

Successive percentage change

\text{net}=a+b+\frac{ab}{100}

Works for two changes in a row, like both dimensions.

Reverse percentage area change

1+\frac{x}{100}=\left(1+\frac{y}{100}\right)^2

Area factor is the side factor squared.

Map areas

\text{true area}=\text{map area}\times(\text{scale})^2

Square the linear scale before converting units.