Geometry — Lines, Angles, Polygons, Triangles & Circles - Cheat sheet
Your last-minute revision companion for geometry fundamentals. Every formula, theorem, and key concept you need to ace UPCAT and other entrance exams.
Lines and Angles
Formulas
Distance = |x₂ - x₁| (horizontal) or |y₂ - y₁| (vertical)
Meaning
Distance between two points on coordinate plane
Watch out
Always use absolute value - distance is never negative
When to use
Finding length of horizontal or vertical line segments
m = (y₂ - y₁)/(x₂ - x₁)
Meaning
m = slope, (x₁,y₁) and (x₂,y₂) are two points
Watch out
Undefined slope when x₂ = x₁ (vertical line)
When to use
Finding slope of line through two points
y - y₁ = m(x - x₁)
Meaning
Point-slope form: m = slope, (x₁,y₁) = known point
Watch out
Don't confuse with slope-intercept form y = mx + b
When to use
Writing equation when you have slope and one point
Important facts
- Sum of angles on a straight line = 180°
- Vertically opposite angles are equal
- Corresponding angles are equal when lines are parallel
- Alternate interior angles are equal when lines are parallel
- Consecutive interior angles are supplementary (180°) when lines are parallel
Key definitions
Parallel Lines
Example
m₁ = m₂ = 2 for lines y = 2x + 1 and y = 2x + 5
Lines with equal slopes that never intersect
Perpendicular Lines
Example
If m₁ = 2, then m₂ = -1/2
Lines whose slopes are negative reciprocals
Vertical Angles
Example
Always equal: ∠1 = ∠3, ∠2 = ∠4
Opposite angles formed by two intersecting lines
Diagrams to know
- Parallel lines cut by transversal showing all 8 angles
- Perpendicular lines forming 90° angles
- Coordinate plane with positive/negative quadrants
Triangles
Formulas
Area = (1/2) × base × height
Meaning
Base = any side, height = perpendicular distance to that side
Watch out
Height must be perpendicular to the base
When to use
Finding triangle area with base and height given
Perimeter = a + b + c
Meaning
a, b, c = lengths of the three sides
Watch out
All three sides must be given
When to use
Finding total distance around triangle
a² + b² = c²
Meaning
Pythagorean theorem: a, b = legs, c = hypotenuse
Watch out
Only works for right triangles (90° angle)
When to use
Right triangles only - finding unknown side
Sum of interior angles = 180°
Meaning
∠A + ∠B + ∠C = 180°
Watch out
Works for all triangles, not just special ones
When to use
Finding unknown angle in any triangle
Common values
Value
1 : √3 : 2
Symbol
short : medium : long
Quantity
30-60-90 triangle sides
Value
1 : 1 : √2
Symbol
leg : leg : hypotenuse
Quantity
45-45-90 triangle sides
Value
3-4-5, 5-12-13, 8-15-17
Symbol
a-b-c
Quantity
Common Pythagorean triples
Important facts
- Triangle inequality: sum of any two sides > third side
- Exterior angle = sum of two non-adjacent interior angles
- In right triangle: side opposite largest angle is longest
- Similar triangles have proportional sides and equal angles
- Altitude creates two right triangles in original triangle
Key definitions
Equilateral Triangle
Example
Side = 5, all angles = 60°
All sides equal, all angles = 60°
Isosceles Triangle
Example
If AB = AC, then ∠B = ∠C
Two sides equal, two angles equal
Right Triangle
Example
3-4-5 triangle where 3² + 4² = 5²
One angle = 90°
Congruent Triangles
Example
SSS, SAS, ASA, AAS prove congruence
Same size and shape - all corresponding parts equal
Diagrams to know
- 30-60-90 triangle with sides in ratio 1:√3:2
- 45-45-90 triangle with sides in ratio 1:1:√2
- Right triangle showing altitude to hypotenuse
Polygons
Formulas
Sum of interior angles = (n-2) × 180°
Meaning
n = number of sides
Watch out
Formula only works for n ≥ 3
When to use
Finding total of all interior angles in any polygon
Each interior angle = (n-2) × 180° ÷ n
Meaning
For regular polygons only (all angles equal)
Watch out
Only for regular polygons - irregular polygons have different angles
When to use
Finding one angle in regular polygon
Number of diagonals = n(n-3)/2
Meaning
n = number of vertices/sides
Watch out
Don't count sides as diagonals
When to use
Counting all possible diagonals in polygon
Sum of exterior angles = 360°
Meaning
Always 360° regardless of number of sides
Watch out
True for all polygons, not just regular ones
When to use
Finding exterior angles or checking work
Important facts
- Triangle: 3 sides, sum of angles = 180°
- Quadrilateral: 4 sides, sum of angles = 360°
- Pentagon: 5 sides, each interior angle = 108° (if regular)
- Hexagon: 6 sides, each interior angle = 120° (if regular)
- Octagon: 8 sides, each interior angle = 135° (if regular)
Key definitions
Regular Polygon
Example
Square, equilateral triangle, regular hexagon
All sides equal and all angles equal
Convex Polygon
Example
Rectangle, pentagon - no sides bend inward
All interior angles less than 180°
Diagonal
Example
In square ABCD, AC and BD are diagonals
Line segment connecting non-adjacent vertices
Diagrams to know
- Regular polygons: triangle through octagon
- Convex vs concave polygon examples
- Polygon with diagonals drawn from one vertex
Quadrilaterals
Formulas
Area of rectangle = length × width
Meaning
l = length, w = width
Watch out
Make sure you're using perpendicular sides
When to use
Finding area of rectangle or square
Area of parallelogram = base × height
Meaning
Height is perpendicular distance between parallel sides
Watch out
Height is NOT the side length - it's perpendicular
When to use
Finding area when you have base and perpendicular height
Area of trapezoid = (1/2)(b₁ + b₂) × h
Meaning
b₁, b₂ = parallel sides, h = height between them
Watch out
Height must be perpendicular to both parallel sides
When to use
Finding area of trapezoid
Area of rhombus = (1/2) × d₁ × d₂
Meaning
d₁, d₂ = lengths of diagonals
Watch out
Diagonals of rhombus are perpendicular
When to use
Finding rhombus area using diagonals
Important facts
- All squares are rectangles and rhombuses
- Diagonals of rectangle are equal and bisect each other
- Diagonals of rhombus are perpendicular and bisect each other
- Opposite angles in parallelogram are equal
- Consecutive angles in parallelogram are supplementary
Key definitions
Square
Example
4 equal sides, 4 right angles
Rectangle with all sides equal
Rectangle
Example
Opposite sides equal and parallel
Parallelogram with 4 right angles
Rhombus
Example
Like a diamond shape, diagonals bisect at right angles
Parallelogram with all sides equal
Parallelogram
Example
Opposite sides equal, opposite angles equal
Quadrilateral with opposite sides parallel
Diagrams to know
- Square with diagonals showing 45° angles
- Rectangle with diagonal creating right triangles
- Parallelogram showing height vs side length
- Trapezoid with height marked perpendicular to bases
Circles
Formulas
Circumference = 2πr = πd
Meaning
r = radius, d = diameter = 2r
Watch out
Use π ≈ 3.14 or leave answer in terms of π
When to use
Finding distance around circle
Area = πr²
Meaning
r = radius
Watch out
Remember to square the radius, not multiply by 2
When to use
Finding space inside circle
Arc length = (θ/360°) × 2πr
Meaning
θ = central angle in degrees, r = radius
Watch out
Angle must be in degrees for this formula
When to use
Finding length of part of circle circumference
Sector area = (θ/360°) × πr²
Meaning
θ = central angle in degrees, r = radius
Watch out
This is area, not arc length
When to use
Finding area of pie-slice portion of circle
Inscribed angle = (1/2) × central angle
Meaning
Angle at circumference is half the central angle
Watch out
Only works for angles with vertex on the circle
When to use
Finding angles inscribed in circles
Common values
Value
≈ 3.14159 or 22/7
Symbol
π
Quantity
π (pi)
Important facts
- Inscribed angle in semicircle is always 90°
- Tangent is perpendicular to radius at point of tangency
- Equal chords are equidistant from center
- Inscribed angles subtending same arc are equal
- Opposite angles of inscribed quadrilateral sum to 180°
Key definitions
Radius
Example
If center is O and point A is on circle, OA is radius
Distance from center to any point on circle
Diameter
Example
Longest possible chord in a circle
Chord that passes through center, equals 2 × radius
Chord
Example
Diameter is the longest chord
Line segment with both endpoints on circle
Tangent
Example
Perpendicular to radius at point of contact
Line that touches circle at exactly one point
Secant
Example
Extended chord becomes secant line
Line that intersects circle at two points
Diagrams to know
- Circle with center, radius, diameter, chord labeled
- Tangent line perpendicular to radius
- Inscribed angle vs central angle relationship
- Circle with inscribed triangle/quadrilateral
Must remember
- Pythagorean theorem: a² + b² = c² (right triangles only)
- Sum of triangle angles = 180°
- Circle area = πr², circumference = 2πr
- Parallel lines have equal slopes, perpendicular lines have negative reciprocal slopes
- Sum of polygon interior angles = (n-2) × 180°
- Inscribed angle = half the central angle
- 30-60-90 triangle sides ratio: 1:√3:2
- 45-45-90 triangle sides ratio: 1:1:√2
- Area formulas: triangle = (1/2)bh, rectangle = lw, trapezoid = (1/2)(b₁+b₂)h
- All exterior angles of any polygon sum to 360°
Last minute tips
- For right triangles, always check if it's a special triangle (30-60-90 or 45-45-90) before using Pythagorean theorem
- When finding polygon angles, write (n-2)×180° first, then divide by n for regular polygons
- Circle problems: if radius doubles, area becomes 4 times larger (r² effect)
- For parallel line problems, identify the transversal first, then use corresponding/alternate angle rules
- Triangle inequality check: each side must be less than sum of other two sides
Comparison tables
Triangle Types by Sides vs Angles
| Classification | Equilateral | Isosceles | Scalene |
|---|---|---|---|
| By sides | All equal | Two equal | All different |
| By angles | All 60° | Two equal | All different |
| Example sides | 5-5-5 | 5-5-7 | 3-4-5 |
Quadrilateral Properties
| Property | Square | Rectangle | Rhombus | Parallelogram |
|---|---|---|---|---|
| All sides equal | Yes | No | Yes | No |
| All angles 90° | Yes | Yes | No | No |
| Diagonals equal | Yes | Yes | No | No |
| Diagonals perpendicular | Yes | No | Yes | No |
Diagrams
Geometry Formula Map
Visual map showing relationships between all major geometry formulas organized by shape type
Triangle Problem Solving Decision Tree
Step-by-step decision process for solving different types of triangle problems
Circle Problem Strategy
Decision tree for choosing the correct circle formula based on what needs to be calculated