50  Geometry

50.1 Introduction

Geometry is the branch of mathematics concerned with the study of shapes, sizes, relative positions of figures, and properties of space.
In aptitude and competitive exams, the focus is on plane geometry (2D figures like triangles, quadrilaterals, circles, polygons) and solid geometry (3D figures like cubes, cylinders, cones, spheres).

This chapter consolidates important theorems, formulas, and properties required for fast problem-solving.

50.2 Basic Geometric Figures

50.2.1 Point, Line, Plane

  • Point: No dimensions, only position.
  • Line: Infinite length, no breadth.
  • Plane: Flat 2D surface extending infinitely.

50.2.2 Angles

  • Acute: \(0^\circ < \theta < 90^\circ\)
  • Right: \(\theta=90^\circ\)
  • Obtuse: \(90^\circ < \theta < 180^\circ\)
  • Straight: \(\theta=180^\circ\)
  • Complementary: Sum = \(90^\circ\)
  • Supplementary: Sum = \(180^\circ\)

50.3 Triangles

50.3.1 Types

  • By sides: equilateral, isosceles, scalene.
  • By angles: acute, right, obtuse.

50.3.2 Properties

  • Sum of interior angles = \(180^\circ\).
  • Exterior angle = sum of two opposite interior angles.
  • In any triangle: side opposite larger angle is longer.

50.3.3 Pythagoras Theorem

For right triangle:
\[ a^2+b^2=c^2 \] (\(c\) = hypotenuse).

50.3.4 Area of Triangle

  1. Using base and height: \(\tfrac{1}{2}bh\).
  2. Heron’s formula: \(\sqrt{s(s-a)(s-b)(s-c)}\), \(s=\tfrac{a+b+c}{2}\).

50.4 Quadrilaterals

50.4.1 Parallelogram

  • Opposite sides parallel and equal.
  • Opposite angles equal, diagonals bisect each other.
  • Area = base × height.

50.4.2 Rectangle

  • Parallelogram with all angles \(90^\circ\).
  • Diagonals equal.
  • Area = \(l\times b\).

50.4.3 Square

  • Rectangle with all sides equal.
  • Area = \(a^2\).

50.4.4 Rhombus

  • Parallelogram with all sides equal.
  • Diagonals bisect at right angles.
  • Area = \(\tfrac{1}{2}d_1d_2\).

50.4.5 Trapezium

  • One pair of opposite sides parallel.
  • Area = \(\tfrac{1}{2}(a+b)h\), \(a,b\) = parallel sides.

50.5 Circle Geometry

50.5.1 Definitions

  • Radius: distance from center to boundary.
  • Diameter = 2 × radius.
  • Chord: line segment joining two points on circle.
  • Tangent: line touching circle at one point.
  • Secant: line cutting circle at two points.

50.5.2 Properties

  • Angle subtended by diameter at circle = \(90^\circ\).
  • Equal chords subtend equal angles at center.
  • Tangent ⟂ radius at point of contact.

50.5.3 Circle Formulas

  • Circumference = \(2\pi r\).
  • Area = \(\pi r^2\).
  • Sector area = \(\frac{\theta}{360^\circ}\pi r^2\).
  • Arc length = \(\frac{\theta}{360^\circ}\cdot 2\pi r\).

50.6 Polygons

  • Polygon with \(n\) sides has sum of interior angles = \((n-2)\times 180^\circ\).
  • Each interior angle (regular polygon) = \(\frac{(n-2)180}{n}\).
  • Number of diagonals = \(\tfrac{n(n-3)}{2}\).

50.7 Solid Geometry (Mensuration – 3D)

50.7.1 Cuboid

  • Surface area = \(2(lb+bh+hl)\).
  • Volume = \(lbh\).

50.7.2 Cube

  • Surface area = \(6a^2\).
  • Volume = \(a^3\).

50.7.3 Cylinder

  • Curved surface area = \(2\pi rh\).
  • Total surface area = \(2\pi r(h+r)\).
  • Volume = \(\pi r^2h\).

50.7.4 Cone

  • Slant height \(l=\sqrt{r^2+h^2}\).
  • CSA = \(\pi rl\).
  • TSA = \(\pi r(l+r)\).
  • Volume = \(\tfrac{1}{3}\pi r^2h\).

50.7.5 Sphere

  • Surface area = \(4\pi r^2\).
  • Volume = \(\tfrac{4}{3}\pi r^3\).

50.8 Coordinate Geometry Applications

  • Equation of circle: \((x-h)^2+(y-k)^2=r^2\).
  • Area of triangle using coordinates:
    \[ \tfrac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)| \]
  • Distance, midpoint, section formulas connect coordinate and geometry concepts.

50.9 Solved Examples

Example 1
Find area of right triangle with base 6 and height 8.
Area = \(\tfrac{1}{2}\times6\times8=24\).

Example 2
Find area of trapezium with parallel sides 8, 12 and height 5.
Area = \(\tfrac{1}{2}(8+12)(5)=50\).

Example 3
Find angle each interior angle of regular hexagon.
\(= \frac{(6-2)180}{6}=120^\circ\).

Example 4
Find volume of cone with \(r=7\), \(h=24\).
Volume = \(\tfrac{1}{3}\pi r^2h=\tfrac{1}{3}\pi(49)(24)=1232\pi\).

Example 5
A chord subtends angle \(90^\circ\) at center of circle of radius 10. Find chord length.
\(=2r\sin(\theta/2)=20\sin 45^\circ=20\cdot\tfrac{\sqrt{2}}{2}=10\sqrt{2}\).

50.10 Practice Problems

  1. Find perimeter and area of equilateral triangle with side 10.
  2. Find length of diagonal of rectangle \(8\times 6\).
  3. Area of rhombus with diagonals 12, 16.
  4. Find surface area and volume of sphere of radius 7.
  5. Find area of regular octagon with side 5 (hint: break into isosceles triangles).
  6. A solid metallic sphere of radius 5 cm is melted to form cones of radius 5 cm and height 10 cm. Find number of cones formed.
  7. In a circle of radius 7 cm, find length of arc subtending \(60^\circ\).

50.11 Summary

  • Geometry covers 2D (plane) and 3D (solid) figures.
  • Key figures: triangles, quadrilaterals, circles, polygons, solids.
  • Properties and formulas should be memorized for speed.
  • Circle theorems, mensuration formulas, and coordinate geometry links are essential.
  • Visualization and formula recall are the keys to mastering geometry in exams.