Geometry Formulas: Complete Formula Sheet

Every geometry formula you are likely to need in one place — area and perimeter, solids, triangles, circles and coordinate geometry. Skim before an exam, or copy a section into the Flashcard Generator to drill it.

Area and perimeter (2D shapes)

Square
Area = s². Perimeter = 4s.
Rectangle
Area = l × w. Perimeter = 2(l + w).
Triangle
Area = ½ × base × height. Perimeter = a + b + c.
Parallelogram
Area = base × perpendicular height.
Trapezium
Area = ½ (a + b) × h, where a and b are the parallel sides.
Rhombus
Area = ½ × d₁ × d₂ (product of the diagonals, halved).
Circle
Area = πr². Circumference = 2πr = πd.
Sector
Area = (θ/360) × πr². Arc length = (θ/360) × 2πr, with θ in degrees.
Regular polygon
Interior angle sum = (n − 2) × 180°. Each exterior angle = 360°/n.

Volume and surface area (3D solids)

Cube
Volume = s³. Surface area = 6s².
Cuboid
Volume = l × w × h. Surface area = 2(lw + lh + wh).
Cylinder
Volume = πr²h. Curved surface = 2πrh. Total surface = 2πr(r + h).
Cone
Volume = ⅓πr²h. Curved surface = πrl, where l is the slant height.
Sphere
Volume = 4/3 πr³. Surface area = 4πr².
Prism
Volume = cross-sectional area × length.
Pyramid
Volume = ⅓ × base area × perpendicular height.

Triangles and trigonometry

Pythagoras' theorem
a² + b² = c², where c is the hypotenuse of a right-angled triangle.
SOH-CAH-TOA
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj.
Sine rule
a/sin A = b/sin B = c/sin C.
Cosine rule
a² = b² + c² − 2bc cos A.
Area of any triangle
Area = ½ ab sin C.
Angle sum
Interior angles of a triangle add to 180°; exterior angle equals the sum of the two opposite interior angles.
Similar triangles
Corresponding angles equal, corresponding sides in a fixed ratio; areas scale by the ratio squared.

Circle theorems

Angle at the centre
The angle at the centre is twice the angle at the circumference on the same arc.
Angle in a semicircle
The angle in a semicircle is always 90°.
Angles in the same segment
Angles subtended by the same arc in the same segment are equal.
Cyclic quadrilateral
Opposite angles of a cyclic quadrilateral sum to 180°.
Tangent–radius
A tangent meets the radius at the point of contact at 90°.
Alternate segment
The angle between a tangent and a chord equals the angle in the alternate segment.

Coordinate geometry

Distance between two points
d = √[(x₂ − x₁)² + (y₂ − y₁)²].
Midpoint
M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
Gradient
m = (y₂ − y₁)/(x₂ − x₁).
Equation of a line
y = mx + c, or y − y₁ = m(x − x₁).
Parallel and perpendicular
Parallel lines share a gradient; perpendicular gradients multiply to −1.
Equation of a circle
(x − a)² + (y − b)² = r², centre (a, b), radius r.

How to use a formula sheet properly

Reading formulas does not make them stick — using them does. Work one question per formula, then write the sheet out from memory and check what is missing. Watch units: areas use square units, volumes cubic units, and mixing centimetres with metres mid-question is the single most common avoidable error in geometry papers.

Frequently asked questions

Which geometry formulas are given in exams?

It varies by board. Sphere and cone formulas are often provided, while area, Pythagoras and the circle theorems usually are not — check your formula booklet and memorise the gaps.

What's the difference between the sine rule and the cosine rule?

Use the sine rule when you have a matching side-and-angle pair. Use the cosine rule when you have three sides, or two sides and the angle between them.

How do I remember circle theorems?

Sketch each one once from memory every day for a week. Six diagrams take two minutes and beat re-reading the list.

Is this formula sheet printable?

Yes — use your browser's print function, or copy sections into the Flashcard Generator to export a PDF deck.

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