Essential Geometry Formulas for A-Level Mathematics

This page collects the most essential geometry formulas and facts for A-Level Mathematics, including triangles, circles, coordinate geometry and vectors.

1. Basic Angle and Shape Facts

Fact Result
Angles on a straight line Sum is 180°
Angles at a point Sum is 360°
Interior angles of a triangle Sum is 180°
Interior angles of a quadrilateral Sum is 360°
Radians and degrees π radians = 180°    therefore   1 radian = 180° / π

2. Right-Angled Triangles and Pythagoras

Result Formula
Pythagoras' theorem a2 + b2 = c2  (c is the hypotenuse)
Area of a right-angled triangle Area = ½ × base × height
Basic trig ratios (right-angled triangle) sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
Relationship between tan, sin and cos tan θ = sin θ / cos θ  (cos θ ≠ 0)

3. General Triangles (Non-Right-Angled)

Result Formula
Sine rule a / sin A = b / sin B = c / sin C
Cosine rule (for side a) a2 = b2 + c2 - 2bc cos A
Cosine rule (for angles) cos A = (b2 + c2 - a2) / (2bc)
Area of a triangle using two sides and included angle Area = ½ ab sin C = ½ bc sin A = ½ ca sin B
Heron's formula for area Let s = (a + b + c) / 2, then Area = √(s(s - a)(s - b)(s - c))

4. Circles and Radian Geometry

Quantity Formula
Circumference of a circle C = 2πr
Area of a circle A = πr2
Arc length (radians) s = rθ  (with θ in radians)
Area of a sector (radians) A = ½ r2θ  (with θ in radians)
Radians to degrees θ° = (θ radians) × 180° / π

5. Coordinate Geometry of Lines

Result Formula
Distance between two points For P(x1, y1), Q(x2, y2):
PQ = √((x2 - x1)2 + (y2 - y1)2)
Midpoint of a line segment Midpoint M = ( (x1 + x2) / 2 , (y1 + y2) / 2 )
Gradient (slope) of a line m = (y2 - y1) / (x2 - x1)  (x2 ≠ x1)
Equation of a line (point-gradient form) y - y1 = m(x - x1)
Gradient of a perpendicular line If one line has gradient m, a perpendicular line has gradient -1/m  (m ≠ 0)
Parallel lines Parallel lines have equal gradients.

6. Coordinate Geometry of Circles

Form Details
Centre-radius form (x - a)2 + (y - b)2 = r2
Centre: (a, b), radius: r
General form x2 + y2 + gx + fy + c = 0
Centre: ( -g/2, -f/2 )
Radius: √((g2 + f2) / 4 - c)
Condition for a point on the circle A point (x, y) lies on the circle if its coordinates satisfy the circle equation.

7. Vectors and Geometry

Result Formula / Description
Magnitude (length) of a vector For a = (a1, a2, a3):
|a| = √(a12 + a22 + a32)
Distance between two points via vectors If points A and B have position vectors a and b, then AB = |b - a|.
Dot product For a = (a1, a2, a3),
    b = (b1, b2, b3):
a · b = a1b1 + a2b2 + a3b3
Dot product and angle between vectors a · b = |a| |b| cos θ
therefore   cos θ = (a · b) / (|a| |b|)
Perpendicular vectors a is perpendicular to b if and only if a · b = 0.
Parallel vectors a is parallel to b if there exists a scalar k such that a = kb.
Vector equation of a line r = a + λd
passes through the point with position vector a in the direction of vector d.

These results cover the core geometry facts and formulas used throughout A-Level Mathematics, especially in trigonometry, coordinate geometry and vectors.

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