Angles
Basic Angle Facts
Angles at a point = 360°
Vertically opposite angles are equal
Angles in Parallel Lines
Alternate angles (Z-angles) are equal.
Corresponding angles (F-angles) are equal.
Co-interior angles (C-angles) add up to 180°.
Angles in Polygons
Each exterior angle of a regular polygon = 360°n
Interior + Exterior = 180°
Worked Example
Find the interior angle of a regular octagon.
Exterior angle = 360 ÷ 8 = 45°
Interior angle = 180 − 45 = 135°
Exam Tip
Always state the angle rule you are using. Marks are awarded for reasons, not just answers!
Properties of Shapes
Quadrilateral Properties
Square: All sides equal, all angles 90°, 4 lines of symmetry.
Rectangle: Opposite sides equal, all angles 90°, 2 lines of symmetry.
Parallelogram: Opposite sides equal and parallel, opposite angles equal.
Rhombus: All sides equal, opposite angles equal, diagonals bisect at right angles.
Trapezium: One pair of parallel sides.
Kite: Two pairs of adjacent equal sides, one line of symmetry.
Congruence & Similarity
Congruent shapes are identical (same shape AND size). Conditions: SSS, SAS, ASA, RHS.
Similar shapes are the same shape but different sizes. Corresponding angles are equal, and sides are in the same ratio (scale factor).
Area scale factor = k²
Volume scale factor = k³
Area & Perimeter
Area Formulas
Triangle: A = 12 × b × h
Parallelogram: A = b × h
Trapezium: A = 12(a + b) × h
Circle: A = πr²
Sector: A = θ360 × πr²
Circumference & Arc Length
Arc length: θ360 × πd
Worked Example
Find the area of a sector with radius 6 cm and angle 120°.
Area = 120360 × π × 6²
= 13 × 36π
= 12π = 37.7 cm² (1 d.p.)
Volume & Surface Area
Volume Formulas
Prism: V = cross-section area × length
Cylinder: V = πr²h
Cone: V = 13πr²h
Sphere: V = 43πr³
Pyramid: V = 13 × base area × h
Surface Area
Cone: SA = πr² + πrl (l = slant height)
Sphere: SA = 4πr²
Key Facts
- Cone and pyramid volumes are ⅓ of the corresponding prism/cylinder
- These formulas are given on the exam formula sheet
Transformations
Four Transformations
Translation: Described by a column vector. Shape stays the same size and orientation.
Reflection: Described by the equation of the mirror line (e.g. x = 2, y = x).
Rotation: Described by centre, angle, and direction.
Enlargement: Described by centre and scale factor. Negative SF = enlargement + rotation 180°. Fractional SF = shape gets smaller.
Common Mistake
"Describe fully" means you must include ALL details. For rotation: centre, angle AND direction. For enlargement: centre AND scale factor.
Pythagoras & Trigonometry
Pythagoras' Theorem
To find the hypotenuse: add the squares, then square root.
To find a shorter side: subtract the squares, then square root.
Worked Example
Find the length of the missing side: hypotenuse = 13, one side = 5.
a² + 5² = 13²
a² = 169 − 25 = 144
a = √144 = 12 cm
Trigonometry (SOHCAHTOA)
1. Label the sides (O, A, H). 2. Choose the right ratio. 3. Substitute and solve.
Sine & Cosine Rules (Higher)
Cosine rule: a² = b² + c² − 2bc cos A
Area = 12ab sin C
Sine rule: Use when you have a pair of opposite side + angle.
Cosine rule: Use when you have 3 sides, or 2 sides + included angle.
Exam Tip
Make sure your calculator is in DEGREES mode, not radians!
Vectors (Higher)
Vector Basics
A vector has magnitude (size) and direction. Written as a column vector or bold letter (e.g. a).
Adding vectors: Add the corresponding components.
Scalar multiplication: Multiply each component by the scalar.
Reverse direction: Negate the vector. If AB = a, then BA = −a.
Vector Proofs
To show points are collinear (on the same line): show one vector is a scalar multiple of another.
To find a midpoint M of AB: OM = OA + 12AB
Circle Theorems (Higher)
The Six Key Theorems
1. The angle at the centre is twice the angle at the circumference.
2. The angle in a semicircle is 90°.
3. Angles in the same segment are equal.
4. Opposite angles of a cyclic quadrilateral add up to 180°.
5. A tangent to a circle is perpendicular to the radius at the point of contact.
6. The alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
Key Facts
- Two tangents from the same external point are equal in length
- The perpendicular from the centre to a chord bisects the chord
- You must state the theorem name in your answer for full marks
Geometry Flashcards
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Geometry - Mock Exam Questions
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