The biggest number that divides exactly into each of the given numbers.
Precisely: The highest common factor (greatest common divisor) of integers is the largest positive integer that divides each of them; by prime factorisation it is the product of the smallest power of each common prime factor.
How it works
Write each number as a product of prime powers. Take only the primes that appear in every number, each with its smallest power, and multiply. Euclid's division algorithm finds it too, by repeated division.
Examples
The HCF of 6 = 2 × 3 and 20 = 2² × 5 is 2.
The HCF of 96 = 2⁵ × 3 and 404 = 2² × 101 is 2² = 4.
The HCF of 6, 72 and 120 is 2 × 3 = 6.
Given that the HCF of 306 and 657 is 9, their LCM is 306 × 657 ÷ 9 = 22,338.
Do not confuse: The HCF is at most the smallest number; the LCM is at least the largest. HCF × LCM equals the product for two numbers only, not for three.
Used in
simplifying fractions, sharing things into equal groups, tiling with the largest square, LCM from HCF
The straight up distance from a base to the opposite corner or side, measured at a right angle to the base.
Precisely: The perpendicular distance from a vertex to the line containing the opposite side (the base) of a triangle, or between the parallel sides of a parallelogram or trapezium.
How it works
Drop a perpendicular from the top corner to the base (extended, if needed): its length is the height. Areas use it: a parallelogram is base × height and a triangle half of that. A slanting side is not the height.
Examples
A parallelogram with base 8 cm and height 5 cm has area 40 cm², however much it leans.
A right triangle with area 54 cm² and one leg 12 cm has the other leg as its height: 2 × 54 ÷ 12 = 9 cm.
A trapezium with parallel sides 40 cm and 20 cm and slant sides 26 cm has height √(26² − 10²) = 24 cm and area 720 cm².
In a triangle with an obtuse angle at the base, the height falls outside the triangle, on the base extended.
Do not confuse: The height must be at a right angle to the base; a slanting side is longer than the height. The height of a triangle is not the median, which goes to the midpoint of the side.
Used in
areas of triangles, parallelograms and trapezia, heights and distances (Class 10)
Half of a sphere, cut through its centre, like a bowl.
Precisely: Either half of a sphere cut by a plane through its centre; its curved surface area is 2πr², its total surface area (with the flat circle) 3πr², and its volume ⅔πr³.
How it works
Halve the sphere's formulas for the curved part and the volume; for the total surface add the flat circular face, πr².
Examples
A hemisphere of radius 2 cm has volume ⅔π × 8 = 16π/3 ≈ 16.76 cm³.
A bowl is a hollow hemisphere; its inside to be painted is 2πr².
The Earth is split into the northern and southern hemispheres by the equator.
Do not confuse: A solid hemisphere has a flat face too, so its total surface area is 3πr², not half of 4πr².
Used in
bowls and domes, combined solids (toys, tops, bird-baths), geography
A way to find the area of a triangle from its three sides alone, with no height needed.
Precisely: For a triangle with sides a, b and c and semi-perimeter s = (a + b + c)/2, area = √(s(s − a)(s − b)(s − c)).
How it works
Find s, half the perimeter. Subtract each side from s, multiply the four numbers s, s − a, s − b and s − c, and take the square root. It is the special case d = 0 of Brahmagupta's formula.
Examples
By Heron's formula, the 3-4-5 triangle has s = 6 and area √(6 × 3 × 2 × 1) = 6.
Sides 7, 24 and 25 cm: s = 28, and Heron's formula gives √(28 × 21 × 4 × 3) = 84 cm², the same as ½ × 7 × 24.
A plot with sides 60, 100 and 140 m has s = 150, so by Heron's formula its area is √(150 × 90 × 50 × 10) = 1500√3 ≈ 2598 m².
For an equilateral triangle of side a, Heron's formula gives (√3/4)a².
Do not confuse: s is half the perimeter, not the whole perimeter. And the formula needs all three sides; two sides alone do not fix the area.
Used in
areas of plots and fields, triangles without a known height, the area of any polygon, cut into triangles
A curve with two separate branches: all the points whose distances from two fixed points (the foci) differ by the same number.
Precisely: The set of all points in a plane the difference of whose distances from two fixed points (the foci) is a constant 2a; with its centre at the origin and foci on the x-axis its equation is x²/a² − y²/b² = 1, where b² = c² − a².
How it works
For each point on it, the distance to one focus minus the distance to the other is always 2a. The points closer to the left focus make the left branch and those closer to the right focus make the right branch. Far out, each branch runs close to two straight lines, its asymptotes.
Examples
The graph of y = 1/x is a hyperbola.
The side of a cooling tower curves like a hyperbola.
The shadow a lampshade throws on a wall has edges shaped like a hyperbola.
x²/16 − y²/9 = 1 is a hyperbola with foci at (5, 0) and (−5, 0).
Do not confuse: A hyperbola is not two parabolas: its branches straighten out along asymptotes, while a parabola keeps bending.
Used in
cooling towers, navigation by radio timing, the paths of fast comets