📏 Math Formulas
31 formulas in 8 sections. For each one you get the formula, what every symbol means, where it is used, an explanation in plain words and a worked example.
Sections: Triangle · Polygon · Pythagoras · Circle · Coordinate Geometry · Straight Line · Angles · Parallel Lines
∠A + ∠B + ∠C = 180°
| ∠A, ∠B, ∠C | the three interior angles |
|---|
Where it is used: To find a missing angle of a triangle when the other two are known — used throughout geometry.
In simple words: The three corners of any triangle always add up to 180 degrees.
Example: If two angles are 60° and 70°, the third = 180−130 = 50°
Exterior angle = sum of the two opposite interior angles
| — | the outside angle at a corner |
|---|
Where it is used: To find an exterior angle of a triangle without knowing the third interior angle.
In simple words: The angle formed outside a corner equals the two inside corners far from it, added together.
Example: Interior 50° and 60° → exterior = 110°
SSS, SAS, ASA, AAS, RHS
| S | side |
|---|---|
| A | angle |
| RHS | right angle-hypotenuse-side |
Where it is used: To prove two triangles are exactly equal in shape and size.
In simple words: If certain matching sides and angles are equal, the triangles are identical.
Example: Two sides and the angle between them equal → SAS
Ratio of areas = (Ratio of sides)²
| — | for two similar triangles |
|---|
Where it is used: To compare areas of similar figures from their side ratio — maps, scale models.
In simple words: If a shape is scaled by 2, its area grows by 2² = 4 times.
Example: Sides in 2:3 → areas in 4:9
A line parallel to one side divides the other two sides proportionally
| — | line parallel to a side of a triangle |
|---|
Where it is used: To find unknown lengths when a line is drawn parallel to a triangle's side.
In simple words: A line parallel to one side cuts the other two sides in the same ratio.
Example: AD/DB = AE/EC
Sum = (n − 2) × 180°
| n | number of sides |
|---|
Where it is used: To find the total of all inside angles of any polygon — tiling, design, construction.
In simple words: Split the shape into triangles: a shape with n sides makes (n−2) triangles, each 180°.
Example: Pentagon (n=5): (5−2)×180 = 540°
Diagonals = n(n − 3) ÷ 2
| n | number of sides |
|---|
Where it is used: To count the diagonals of any polygon.
In simple words: A quick count of how many corner-to-corner lines a shape has.
Example: Hexagon (n=6) → 6×3/2 = 9 diagonals
a² + b² = c²
| c | hypotenuse (longest side, opposite right angle) |
|---|---|
| a, b | the two shorter sides |
Where it is used: In a right-angled triangle, to find one side from the other two — construction, navigation, distances.
In simple words: In an 'L'-shaped corner, the slanted side squared equals the two straight sides squared and added.
Example: a=3, b=4 → c = √(9+16) = 5
∠ in semicircle = 90°
| ∠ | angle at the circle from a diameter |
|---|
Where it is used: A circle theorem used to prove right angles and solve geometry problems.
In simple words: Any angle drawn from the ends of a diameter to a point on the circle is a perfect right angle.
Example: A triangle drawn on a diameter always has a 90° corner.
Angle at centre = 2 × angle at the circumference
| — | angles standing on the same arc |
|---|
Where it is used: A circle theorem used to find unknown angles in circle geometry.
In simple words: An angle made at the centre is always double the same angle made at the edge, on the same arc.
Example: Angle at edge 40° → angle at centre 80°
A tangent is perpendicular (90°) to the radius at the point of contact
| — | tangent touching the circle |
|---|
Where it is used: A circle theorem used to find angles and lengths involving tangents.
In simple words: Where a line just touches a circle, it meets the radius at a right angle.
Example: Radius to the touch point makes 90° with the tangent
Two tangents from an outside point are equal in length
| — | tangents from one external point |
|---|
Where it is used: To find tangent lengths and prove equalities in circle geometry.
In simple words: From a point outside a circle, both touching lines are the same length.
Example: PA = PB for tangents from P
Opposite angles add to 180°
| — | a four-sided figure inscribed in a circle |
|---|
Where it is used: To find angles of a quadrilateral whose corners lie on a circle.
In simple words: In a four-sided shape drawn inside a circle, opposite corners add to 180°.
Example: If one angle is 110°, the opposite is 70°
d = √((x₂−x₁)² + (y₂−y₁)²)
| (x₁, y₁) | first point |
|---|---|
| (x₂, y₂) | second point |
Where it is used: To find the straight-line distance between two points on a graph — maps, navigation, physics.
In simple words: It's Pythagoras on a graph: the gap sideways and the gap up, squared, added and rooted.
Example: (0,0) to (3,4): d = √(9+16) = 5
M = ((x₁+x₂)/2, (y₁+y₂)/2)
| (x₁,y₁), (x₂,y₂) | the two end points |
|---|
Where it is used: To find the exact centre point between two points — used in geometry and graphics.
In simple words: Average the two x-values and the two y-values to land exactly in the middle.
Example: Midpoint of (2,4) and (6,8) = (4,6)
P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))
| m : n | the dividing ratio |
|---|---|
| (x₁,y₁), (x₂,y₂) | the two points |
Where it is used: To find a point that divides a line segment in a given ratio — used in geometry and computer graphics.
In simple words: It finds a point that splits the line between two points in a chosen ratio, not just the middle.
Example: (2,3),(8,9) in 1:2 → P = (4,5)
m = (y₂ − y₁) ÷ (x₂ − x₁)
| (x₁,y₁), (x₂,y₂) | two points on the line |
|---|---|
| m | the slope |
Where it is used: To measure how steep a line is — graphs, ramps, rates of change in science and economics.
In simple words: It's the rise divided by the run: how much the line goes up for every step across.
Example: (1,2) to (3,6): m = (6−2)/(3−1) = 2
A = ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
| (x₁,y₁),(x₂,y₂),(x₃,y₃) | the three vertices |
|---|
Where it is used: To find the area of a triangle from the coordinates of its corners — surveying, graphics.
In simple words: Plug the three corner points into this pattern to get the area, no height needed.
Example: (0,0),(4,0),(0,3): A = ½|0+12+0| = 6
G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
| (x₁,y₁),(x₂,y₂),(x₃,y₃) | the three vertices |
|---|
Where it is used: To find the balancing point (centroid) of a triangle from its corner coordinates.
In simple words: Average the three corner points to find the triangle's balance point.
Example: (0,0),(6,0),(0,6) → G = (2,2)
Points are collinear if the triangle area = 0
| — | three points on a graph |
|---|
Where it is used: To check whether three points lie on the same straight line.
In simple words: If the 'triangle' they form has zero area, the three points are in a line.
Example: (1,1),(2,2),(3,3) give area 0 → collinear
y = mx + c
| m | slope (steepness) |
|---|---|
| c | y-intercept (where it crosses the y-axis) |
Where it is used: The standard equation of a straight line — graphing, linear relationships, trend lines.
In simple words: 'm' tells how steep the line is and 'c' tells where it starts on the up-axis.
Example: y = 2x + 1 has slope 2 and crosses y-axis at 1
y − y₁ = m(x − x₁)
| m | slope |
|---|---|
| (x₁,y₁) | a known point on the line |
Where it is used: To write the equation of a line when a point on it and its slope are known.
In simple words: Start from a known point and use the slope to describe every other point on the line.
Example: Point (1,2), slope 3 → y − 2 = 3(x − 1)
tan θ = |(m₁ − m₂) ÷ (1 + m₁m₂)|
| m₁, m₂ | the slopes of the two lines |
|---|
Where it is used: To find the angle at which two lines cross.
In simple words: Uses the two slopes to find the crossing angle.
Example: m₁=1, m₂=0 → tanθ = 1, so θ = 45°
Parallel: m₁ = m₂; Perpendicular: m₁ × m₂ = −1
| m₁, m₂ | the slopes |
|---|
Where it is used: To test if two lines are parallel or at right angles from their slopes.
In simple words: Equal slopes mean parallel; slopes multiplying to −1 mean a right angle.
Example: Slopes 2 and −½ → perpendicular
d = |ax₁ + by₁ + c| ÷ √(a² + b²)
| ax+by+c=0 | the line |
|---|---|
| (x₁, y₁) | the point |
Where it is used: To find the shortest (perpendicular) distance from a point to a line.
In simple words: Gives the straight-line gap between a point and a line.
Example: Point (0,0) to 3x+4y−10=0 → 10/5 = 2
x/a + y/b = 1
| a | x-intercept |
|---|---|
| b | y-intercept |
Where it is used: To write a line's equation from where it cuts the two axes.
In simple words: Uses the two points where the line crosses the axes.
Example: Cuts axes at 2 and 3 → x/2 + y/3 = 1
Complementary: A + B = 90° · Supplementary: A + B = 180°
| A, B | the two angles |
|---|
Where it is used: To find a missing angle that pairs with another to make a right angle or a straight line.
In simple words: Two angles that add to 90° are complementary; two that add to 180° are supplementary.
Example: Complement of 30° = 60°; supplement of 30° = 150°
Angles on a line add to 180°
| — | adjacent angles on a straight line |
|---|
Where it is used: To find a missing angle that sits on a straight line with another.
In simple words: A straight line makes a half turn, so angles along it add to 180°.
Example: If one angle is 120°, the other is 60°
Angles around a point add to 360°
| — | angles meeting at one point |
|---|
Where it is used: To find a missing angle among several meeting at a point.
In simple words: A full turn around a point is 360°, so all the angles there add to 360°.
Example: Three angles 100°+150°+? = 360° → ? = 110°
Vertically opposite angles are equal
| — | angles opposite each other where two lines cross |
|---|
Where it is used: To find equal angles formed when two lines cross.
In simple words: When two lines cross, the angles directly across from each other are equal.
Example: One angle 70° → its opposite is also 70°
Corresponding = equal, Alternate = equal, Co-interior add to 180°
| — | angles made by a line crossing two parallel lines |
|---|
Where it is used: To find angles when a transversal cuts two parallel lines.
In simple words: Parallel lines make matching angles equal and same-side inside angles add to 180°.
Example: Alternate angles are always equal