Geometry Basics – Quick Study Notes
(For JNU Entrance – Quantitative Aptitude)
1. Simple Explanation
Geometry is the branch of mathematics that deals with points, lines, angles, shapes, and their properties. In an entrance exam you’ll encounter problems about:
- Basic elements – point, line, plane, segment, ray
- Angle types – acute, right, obtuse, straight, reflex
- Polygons – triangles, quadrilaterals, pentagon, etc.
- Circle – radius, diameter, chord, arc, sector
- Area & Perimeter – formulas for common shapes
- Coordinate geometry basics – distance, midpoint, slope (optional)
2. Key Points
| Concept | Definition | Typical Formula / Property |
|---|---|---|
| Point | Location in space, no size | – |
| Line | Infinite points in one dimension | – |
| Line Segment | Two points with finite length | – |
| Ray | One end point & extends infinitely | – |
| Angle | Two rays sharing a vertex | Angle = α + β (sum of adjacent angles) |
| Types of Angles | Acute <90°, Right =90°, Obtuse >90° & <180°, Straight =180°, Reflex >180° |
– |
| Triangle | 3 sides, 3 angles | a + b + c = 180° |
| Right Triangle | One right angle | Pythagoras: a² + b² = c² |
| Isosceles Triangle | Two equal sides | Base angles equal |
| Equilateral Triangle | All sides equal | All angles 60° |
| Quadrilateral | 4 sides | Sum of interior angles = 360° |
| Rectangle | Opposite sides equal, all right angles | P = 2(l + w) , A = l × w |
| Square | All sides equal, all right angles | P = 4a , A = a² |
| Parallelogram | Opposite sides parallel & equal | P = 2(a + b) , A = b × h |
| Rhombus | All sides equal | Diagonals perpendicular, bisect angles |
| Circle | Set of points at equal distance (radius) from centre | C = 2πr , A = πr² |
| Chord | Line segment with both endpoints on circle | Chord = 2r sin(θ/2) (θ = subtended angle) |
| Arc | Part of circumference | Arc length = rθ (θ in radians) |
| Sector | Region bounded by two radii & an arc | Area = (θ/360°) × πr² |
| Distance (2‑D) | Between (x₁,y₁) & (x₂,y₂) | d = √[(x₂‑x₁)² + (y₂‑y₁)²] |
| Midpoint | Mid of a segment | M = ((x₁+x₂)/2 , (y₁+y₂)/2) |
| Slope | Rise ÷ Run | m = (y₂‑y₁)/(x₂‑x₁) |
Tip: For exam speed, memorize the most common formulas:
- Triangle area:
½ × base × height- Rectangle area/perimeter:
l×w,2(l+w)- Circle area/perimeter:
πr²,2πr- Pythagoras for right triangles.
3. Worked Example
Problem
A right‑angled triangle has legs of 6 cm and 8 cm. Find:
- Hypotenuse
- Area
- Perimeter
Solution
-
Hypotenuse (c) – Use Pythagoras:
[ c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm} ] -
Area (A) – Right triangle area:
[ A = \frac{1}{2} \times 6 \times 8 = \frac{1}{2} \times 48 = 24 \text{ cm}^2 ] -
Perimeter (P) – Sum of all sides:
[ P = 6 + 8 + 10 = 24 \text{ cm} ]
Answer
- Hypotenuse = 10 cm
- Area = 24 cm²
- Perimeter = 24 cm
4. Quick Revision Summary
- Angles – Acute, right, obtuse, straight, reflex.
- Triangle –
a + b + c = 180°; right triangle → Pythagoras. - Quadrilateral – Sum of interior angles
360°. - Rectangle/Square – All right angles.
- Parallelogram/Rhombus – Opposite sides equal & parallel.
- Circle –
C = 2πr,A = πr²; chord & arc properties. - Area formulas –