Garden
Seeding a lawn on a 12 × 8 m plot covers 96 m², and the bag calls for 35 g per square metre. That comes to 3.36 kg, so one 5 kg bag does the job and still leaves enough to patch the bare spots.
Area of a rectangle Convert it here: Area
Sailing
A mainsail with a 4.2 m boom and a 12 m luff has an area of ½ · 4.2 · 12 = 25.2 m². The sailmaker prices the cloth by that number and a racing class caps it — two sails of equal area may differ in cut, never in size.
Area and perimeter of a triangle
Sewing
A round table top 120 cm across covers π · 0.6² = 1.13 m², while a cloth with a 25 cm drop is a circle 170 cm across — π · 0.85² = 2.27 m². A quarter of a metre of overhang doubles the fabric.
Area and circumference of a circle
Ladder
A step ladder with both legs at 75° to the floor opens at the top to 180° − 75° − 75° = 30°. Set more shallowly, at 70° each, it opens to 40°: steadier on the floor, shorter in reach.
Angles
Television
A 55-inch television gives 139.7 cm of diagonal, not of width. The 16 : 9 ratio together with the theorem yields 121.8 cm across and 68.5 cm high — those are the numbers to hold against the shelf.
The Pythagorean theorem Convert it here: Length
Aquarium
A tank of 100 × 40 × 50 cm holds 200,000 cm³, that is 200 l — and the same 200 kg of water before gravel and rocks go in. The cabinet under it has to carry a quarter of a tonne.
Volume of solids Convert it here: Volume
Wrapping
A box of 30 × 20 × 15 cm has 2 · (600 + 300 + 450) = 2,700 cm² of surface, that is 0.27 m². A roll of paper 70 cm × 2 m offers 1.4 m², enough for five such boxes — four once the overlaps are counted.
Surface area of solids
Kitchen
A pot 24 cm across filled to 12 cm holds π · 12² · 12 = 5,429 cm³, that is 5.4 l. Soup for twenty at 300 ml a bowl needs 6 l — and that surplus ends up on the hob.
Volume units Convert it here: Volume
Screens, windows and the centre of a button
Every pixel on a screen has an address in a coordinate system. A button with corners (320, 180) and (480, 240) has its centre at (400, 210) — the average of the two corners. An interface that "centres itself" is computing exactly that formula, and the width of the button is 480 − 320 = 160 pixels, which is a subtraction of abscissas.
The coordinate plane
Roofs and roof planes
A hipped roof plane is a trapezoid with parallel sides of 12 m and 8 m and a height of 5 m. Its area is (12 + 8) · 5 / 2 = 50 m², so at 7 tiles per square metre one plane needs 350 tiles ordered.
Quadrilaterals Convert it here: Area
Joinery and a 45° mitre
A picture frame made of mouldings cut at 45° closes each corner with two 45°–45°–90° triangles. With a moulding 4 cm wide the mitre is 4√2 ≈ 5.66 cm long, so four corners add about 22.6 cm of material on top of the sum of the picture sides.
Congruence and special triangles
The height of a tree from the length of a shadow
A 1.5 m pole casts a 2 m shadow while the tree beside it casts an 18 m one. The triangles are similar, so the tree is 1.5 · 18 / 2 = 13.5 m tall. The whole measurement comes down to one proportion and a tape measure.
Similarity and the intercept theorem
A clock and the angle between the hands
A clock face is a circle cut into 12 equal arcs of 30°. At 4 o’clock the hands are separated by a central angle of 4 · 30° = 120°, and on a face of radius 12 cm the arc between them measures 2π · 12 · 120 / 360 ≈ 25.1 cm.
The circle: angles, tangent, regular polygons
2D graphics and moving a sprite
A game character drawn at (120, 80) is to move by the vector [45, −20]. The engine adds the coordinates and draws it at (165, 60). The same operation run backwards — from (165, 60) to the starting point — is a subtraction, and that is what an undo is built on.
Geometric transformations
A box and the longest thing that fits in it
A carton with inside dimensions 60 × 40 × 30 cm takes an object longer than any of its edges: along the diagonal it fits √(60² + 40² + 30²) = √6100 ≈ 78 cm. That is the only number to compare with the length of a fishing rod, an easel or a length of trim before packing.
Solid geometry: angles, segments and sections
Navigating a kilometre grid
Two points on a map have coordinates (12, 5) and (20, 11) in kilometres. The straight-line distance is √(8² + 6²) = 10 km — the number a pilot or a rescue team compares with their range before working out a route by road.
Analytic geometry
Navigation and a wind correction
An aircraft flies at [180, 0] km/h relative to the air while the wind carries it by [0, 60] km/h. Its speed over the ground is the sum [180, 60], that is √(180² + 60²) ≈ 190 km/h on a heading offset from the course — and that offset is what the pilot corrects for.
Vectors in the plane

Branch formulas

Branch: Geometry

Area of a rectangle

Area of a rectangle
$A = a \cdot b$
a and b are the lengths of two adjacent sides
Area of a square
$A = a^2$
a square is a rectangle with equal sides
Perimeter of a rectangle
$P = 2a + 2b = 2(a + b)$
the total length of all four sides
A side from the area
$a = \frac{A}{b}, \quad b = \frac{A}{a}$
the area formula reversed — division instead of multiplication
Units of area
$1\,\text{m}^2 = 10\,000\,\text{cm}^2$
the side grows 100×, the area 100 · 100×

Area and perimeter of a triangle

Area of a triangle
$A = \frac{1}{2} \cdot a \cdot h$
a is the base, h is the height dropped onto that base
Perimeter of a triangle
$P = a + b + c$
the sum of the three side lengths
Area of a right triangle
$A = \frac{1}{2} \cdot a \cdot b$
the legs are each other’s base and height
Height from the area
$h = \frac{2A}{a}$
the area formula, the other way round
The triangle inequality
$|a - b| < c < a + b$
not every three segments make a triangle

Area and circumference of a circle

Area of a circle
$A = \pi r^2$
r is the radius
Circumference
$C = 2\pi r = \pi d$
d = 2r, so the two spellings mean the same thing
Diameter and radius
$d = 2r$
the diameter is twice the radius
The number π
$\pi = \frac{C}{d} \approx 3.14$
the same ratio for every circle — which is why it has a name
Radius from the area
$r = \sqrt{\frac{A}{\pi}}$
the area formula, the other way round

Angles

Angle sum of a triangle
$\alpha + \beta + \gamma = 180^\circ$
always, for every triangle in the plane
Complementary angles
$\alpha + \beta = 90^\circ$
together they make a right angle
Supplementary angles
$\alpha + \beta = 180^\circ$
together they make a straight line
Acute angle
$0^\circ < \alpha < 90^\circ$
a right angle is 90°, an obtuse one is 90° < α < 180°
The third angle of a triangle
$\gamma = 180^\circ - \alpha - \beta$
two angles are enough to find the third
Corresponding angles
$\alpha = \beta$
at two parallel lines cut by a transversal
Alternate angles
$\alpha = \gamma$
between the parallels, on opposite sides of the transversal
Co-interior angles
$\alpha + \delta = 180^\circ$
the one pair of the three that is supplementary rather than equal
Angle sum of a polygon
$S_n = (n - 2) \cdot 180^\circ$
n − 2 triangles, 180° each

The Pythagorean theorem

The Pythagorean theorem
$a^2 + b^2 = c^2$
a and b are the legs, c is the hypotenuse
The hypotenuse
$c = \sqrt{a^2 + b^2}$
the longest side, opposite the right angle
A leg
$b = \sqrt{c^2 - a^2}$
the theorem the other way round — subtraction, not addition
Diagonal of a square
$d = a\sqrt{2}$
the theorem applied to a triangle with legs a and a
The 3-4-5 triple
$3^2 + 4^2 = 5^2$
the next ones: 5-12-13, 8-15-17, 7-24-25

Volume of solids

Cuboid
$V = a \cdot b \cdot c$
a, b, c are the three edges meeting at one vertex
Cube
$V = a^3$
a cuboid with all edges equal
Prism and cylinder
$V = A_b \cdot H$
base area times height — one formula for both solids
Cylinder
$V = \pi r^2 h$
the base is a circle, so A_b = πr²
Pyramid and cone
$V = \frac{1}{3} \cdot A_b \cdot H$
same base and height as the prism, but three times less inside
Cone
$V = \frac{1}{3} \pi r^2 h$
one third of the cylinder on the same base and height
Sphere
$V = \frac{4}{3} \pi r^3$
the only formula here with the radius to the third power

Surface area of solids

The general rule
$A = A_b + A_l$
base area plus lateral area
Cuboid
$A = 2(ab + bc + ac)$
three pairs of identical faces
Cube
$A = 6a^2$
six identical squares
Cylinder
$A = 2\pi r^2 + 2\pi rh = 2\pi r(r + h)$
two circles and a rectangle with sides 2πr and h
Pyramid on a square base
$A = a^2 + 2al$
l is the slant height of a face, not the height of the solid
Cone
$A = \pi r^2 + \pi rl = \pi r(r + l)$
l is the slant height — from the apex to the rim of the base
Slant height of a cone
$l = \sqrt{r^2 + h^2}$
from the Pythagorean theorem for the radius and the height
Sphere
$A = 4\pi r^2$
exactly four areas of a great circle

Volume units

Cubic centimetre
$1\,\text{cm}^3 = 1\,\text{ml}$
one volume, two names
A cubic decimetre is a litre
$1\,\text{dm}^3 = 1\,\text{l} = 1000\,\text{cm}^3$
a cube of edge 10 cm
Litre and millilitre
$1\,\text{l} = 1000\,\text{ml}$
the same thousand, written differently
Cubic metre
$1\,\text{m}^3 = 1000\,\text{dm}^3 = 1000\,\text{l}$
a cube of edge 1 m
Metres to centimetres
$1\,\text{m}^3 = 1\,000\,000\,\text{cm}^3$
100³, because 1 m = 100 cm
The scaling rule
$k \text{ times the edge} \Rightarrow k^3 \text{ times the volume}$
the length factor is raised to the third power

The coordinate plane

Coordinates of a point
$A = (x_A,\ y_A)$
the abscissa first, then the ordinate
Signs in the quadrants
$\text{I}: (+,\, +), \quad \text{II}: (-,\, +), \quad \text{III}: (-,\, -), \quad \text{IV}: (+,\, -)$
numbered counter-clockwise
Segment parallel to the x axis
$|AB| = |x_B - x_A|$
both ordinates are equal
Segment parallel to the y axis
$|AB| = |y_B - y_A|$
both abscissas are equal
Midpoint of a segment
$S = \left( \frac{x_A + x_B}{2},\ \frac{y_A + y_B}{2} \right)$
the average of the endpoints, coordinate by coordinate
The other endpoint
$B = (2x_S - x_A,\ 2y_S - y_A)$
the same formula read backwards

Quadrilaterals

Area of a trapezoid
$P = \frac{(a + b) \cdot h}{2}$
a and b are the parallel sides, h the height between them
Area of a parallelogram
$P = a \cdot h$
h is the altitude onto side a, not the neighbouring side
Area of a rhombus from its diagonals
$P = \frac{e \cdot f}{2}$
e and f are the diagonals, always perpendicular
Angle sum of a quadrilateral
$\alpha + \beta + \gamma + \delta = 360^\circ$
two triangles of 180° each
Angles on the same leg
$\alpha + \beta = 180^\circ$
in a trapezoid and in a parallelogram
A base of a trapezoid from its area
$b = \frac{2P}{h} - a$
the area formula read backwards

Congruence and special triangles

Congruent triangles
$\triangle ABC \cong \triangle DEF$
same shape and same size
The 45°–45°–90° triangle
$a : a : a\sqrt{2}$
half of a square of side a
The 30°–60°–90° triangle
$a : a\sqrt{3} : 2a$
half of an equilateral triangle of side 2a
Height of an equilateral triangle
$h = \frac{a\sqrt{3}}{2}$
the short leg is a/2, the hypotenuse a
Area of an equilateral triangle
$P = \frac{a^2\sqrt{3}}{4}$
base a times the height, halved

Similarity and the intercept theorem

Scale factor
$k = \frac{a'}{a}$
the image side divided by the original side
The intercept theorem
$\frac{|SA|}{|SC|} = \frac{|AB|}{|CD|}$
segments on the arms of an angle cut by parallels
Perimeter of a similar figure
$L' = k \cdot L$
every length grows k times
Area of a similar figure
$P' = k^2 \cdot P$
an area depends on two dimensions
Volume of a similar solid
$V' = k^3 \cdot V$
a volume depends on three dimensions

The circle: angles, tangent, regular polygons

Central and inscribed angle
$\alpha = 2\beta$
both standing on the same arc
Angle over a diameter
$\beta = 90^\circ \iff AB \ \text{is a diameter}$
Thales's theorem on the circle
Tangent segment
$t^2 = d^2 - r^2$
d is the distance from the point to the centre
Chord and its distance from the centre
$\left(\frac{c}{2}\right)^2 + h^2 = r^2$
the perpendicular from the centre bisects the chord
Interior angle of a regular polygon
$\alpha_n = \frac{(n - 2) \cdot 180^\circ}{n}$
the central angle is 360°/n
Arc length
$l = 2\pi r \cdot \frac{\alpha}{360^\circ}$
the fraction of the circumference set by the angle
Area of a sector
$P = \pi r^2 \cdot \frac{\alpha}{360^\circ}$
the same fraction, of the area this time

Geometric transformations

Reflection in the x axis
$S_{OX}: (x,\ y) \mapsto (x,\ -y)$
the ordinate changes sign
Reflection in the y axis
$S_{OY}: (x,\ y) \mapsto (-x,\ y)$
the abscissa changes sign
Reflection in the origin
$S_{O}: (x,\ y) \mapsto (-x,\ -y)$
the same thing as a 180° rotation
Reflection in a point S(p, q)
$S_{S}: (x,\ y) \mapsto (2p - x,\ 2q - y)$
S is the midpoint of AA′
Translation by the vector [a, b]
$T_{\vec{v}}: (x,\ y) \mapsto (x + a,\ y + b)$
read backwards it is a subtraction

Solid geometry: angles, segments and sections

Diagonal of a cuboid
$D^2 = a^2 + b^2 + H^2$
a and b are the base edges, H the height
Diagonal of a cube
$D = a\sqrt{3}$
the case a = b = c
Angle between the diagonal and the base
$\tan \alpha = \frac{H}{d}$
d is the diagonal of the base
Apothem of a pyramid
$m^2 = H^2 + \left(\frac{a}{2}\right)^2$
it stands over half a base edge
Lateral edge of a pyramid
$b^2 = H^2 + \left(\frac{a\sqrt{2}}{2}\right)^2$
it stands over half the base diagonal
Diagonal section of a cube
$P = a^2\sqrt{2}$
a rectangle with sides a and a√2

Analytic geometry

Distance between two points
$|AB| = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}$
the Pythagorean theorem on coordinate differences
Slope
$a = \frac{y_B - y_A}{x_B - x_A}$
the rise divided by the run
Slope-intercept form
$y = ax + b$
b is where the line crosses the y axis
General form of a line
$Ax + By + C = 0$
covers vertical lines too, with B = 0
Condition for parallel lines
$a_1 = a_2 \iff k \parallel l$
equal slopes
Condition for perpendicular lines
$a_1 \cdot a_2 = -1 \iff k \perp l$
the slope flipped over and negated
Equation of a circle
$(x - a)^2 + (y - b)^2 = r^2$
centre S(a, b), radius r

Vectors in the plane

Coordinates of a vector
$\vec{AB} = [x_B - x_A,\ y_B - y_A]$
the end minus the start
Length of a vector
$|\vec{v}| = \sqrt{x^2 + y^2}$
the Pythagorean theorem on the coordinates
Sum of two vectors
$\vec{u} + \vec{v} = [u_x + v_x,\ u_y + v_y]$
add the matching coordinates
Difference of two vectors
$\vec{u} - \vec{v} = [u_x - v_x,\ u_y - v_y]$
the same as adding the opposite vector
Multiplication by a number
$k \cdot \vec{v} = [k x,\ k y]$
a negative k reverses the sense
Length after scaling
$|k \cdot \vec{v}| = |k| \cdot |\vec{v}|$
scales by the absolute value

A luminous right triangle with squares built on its sides drawn as a constellation, beside a circle with its radius, against a cosmic nebula in violet and magenta

Geometry

Why we measure area in squares

Perimeter is counted in metres, area in square metres — and that single exponent is behind most of the mistakes people make in plane geometry. It is why 1 m² is not 100 cm², why two plots with identical area can need wildly different amounts of fencing, and why the Pythagorean theorem is really a statement about the areas of squares rather than the lengths of sides.

11 min