Geometry: Notes और MCQ Practice

आसान भाषा में:

Geometry में points, lines, angles, circles वगैरह के basic properties को याद रखना है. ये सब रोज़ के सवालों में बार‑बार आते हैं, तो याद रखो: distance formula, area formulas, और basic angle rules.

Important Formulas / Rules

• Distance between (x1, y1) और (x2, y2) = √[(x2‑x1)² + (y2‑y1)²]
• Area of Triangle = ½ × base × height
• Area of Circle = π × r²
• Perimeter of Rectangle = 2 × (length + breadth)
• Angle sum of triangle = 180°
• Pythagoras: a² + b² = c² (right‑angled triangle)

Quick Tricks

1. Half‑product rule – अगर दो sides known हों और included angle known हो, तो area = ½ × side1 × side2 × sin(included angle).
2. Circle chord trick – यदि chord का distance from centre ‘d’ और radius ‘r’ है, तो chord length = 2 × √(r² − d²).

Common Mistakes

1. Radius को diameter समझ लेना; हमेशा याद रखो diameter = 2 × radius.
2. Angle units mix‑up – जब sin, cos use करो तो angle को degree में रखो, नहीं तो radian conversion भूल जाओ.

1. Concept

Geometry deals with the properties and relationships of points, lines, angles, surfaces, and solids. In SSC CGL the focus is on 2‑D figures (triangles, quadrilaterals, circles) and basic 3‑D solids (cube, cuboid, sphere). Questions test your ability to calculate perimeter, area, volume, and to use similarity, Pythagoras theorem, and basic coordinate geometry.

2. Important formulas / rules / facts

• Perimeter = sum of all sides.
• Area of a triangle = ½ × base × height.
• Heron’s formula: s = (a+b+c) ÷ 2, Area = √[s(s‑a)(s‑b)(s‑c)].
• Pythagoras theorem: In a right‑angled triangle, a² + b² = c² (c is the hypotenuse).
• Area of a quadrilateral:
• Rectangle = length × breadth.
• Square = side².
• Parallelogram = base × height.
• Trapezium = ½ × (sum of parallel sides) × height.
• Circle:
• Circumference = 2 × π × radius.
• Area = π × radius².
• Volume:
• Cube = side³.
• Cuboid = length × breadth × height.
• Cylinder = π × radius² × height.
• Sphere = 4 ÷ 3 × π × radius³.
• Similarity: Corresponding sides are in the same ratio; area ratio = (side ratio)².
• Coordinate geometry: Distance between (x1,y1) and (x2,y2) = √[(x2‑x1)² + (y2‑y1)²]; Slope = (y2‑y1) ÷ (x2‑x1).

3. Solved examples (exam style)

Example 1

A right‑angled triangle has legs of 6 cm and 8 cm. Find its hypotenuse and area.

Solution

• Hypotenuse (c) = √(6² + 8²) = √(36+64) = √100 = 10 cm.
• Area = ½ × 6 × 8 = 24 cm².

Example 2

The perimeter of a square is 48 cm. Find its diagonal.

Solution

• Side = Perimeter ÷ 4 = 48 ÷ 4 = 12 cm.
• Diagonal = side × √2 = 12 × 1.414 ≈ 16.97 cm.

Example 3

A circle and a square have the same perimeter. If the side of the square is 10 cm, find the area of the circle (take π = 22/7).

Solution

• Perimeter of square = 4 × 10 = 40 cm → this is also the circumference of the circle.
• Circumference = 2 × π × r ⇒ r = Circumference ÷ (2π) = 40 ÷ (2 × 22/7) = 40 ÷ (44/7) = 40 × 7 ÷ 44 = 280 ÷ 44 = 6.36 cm (approx).
• Area of circle = π × r² = 22/7 × (6.36)² ≈ 22/7 × 40.45 ≈ 127.3 cm².

Example 4

A cuboid has length 12 cm, breadth 5 cm and height 4 cm. Find its total surface area and volume.

Solution

• Surface area = 2(lb + bh + lh) = 2(12×5 + 5×4 + 12×4) = 2(60 + 20 + 48) = 2×128 = 256 cm².
• Volume = l × b × h = 12 × 5 × 4 = 240 cm³.

Example 5

In a triangle, two sides are 7 cm and 9 cm and the included angle is 60°. Find the area.

Solution

• Area = ½ × a × b × sin(C) = ½ × 7 × 9 × sin60°.
• sin60° = √3 ÷ 2 ≈ 0.866.
• Area = 0.5 × 7 × 9 × 0.866 ≈ 27.2 cm².

4. Shortcuts and tricks

• Half‑product rule: For any triangle, Area = ½ × base × height – remember the “half” is easy to miss.
• Quick diagonal of square: Multiply side by 1.414 (≈√2) – no need to calculate √2 each time.
• Perimeter‑to‑radius conversion: If circumference is given, radius = circumference ÷ (2π). Keep π as 22/7 for fast mental division.
• Similarity shortcut: When two similar figures are given, find the linear ratio first; then square it for area ratio.
• Coordinate distance: Use the “difference of squares” trick – (x2‑x1)² and (y2‑y1)² are often small numbers, square them quickly.

5. Common mistakes to avoid

• Forgetting the “½” in triangle area formulas.
• Mixing up radius and diameter in circle formulas.
• Using π = 3.14 in a question that expects 22/7 (or vice‑versa) – check the instruction.
• Applying Pythagoras to a non‑right‑angled triangle.
• Ignoring units; always write the final answer with correct unit (cm, cm², cm³).