Mensuration, Geometry, and Trigonometry

Mensuration

Mensuration is a branch of mathematics that deals with the measurement of geometric figures. It involves calculating lengths, areas, volumes, and perimeters of various shapes. For competitive exams, understanding the formulas and their applications is crucial. We will cover 2D (plane figures) and 3D (solid figures) mensuration.

2D Mensuration (Plane Figures)

1. Square

A square is a quadrilateral with four equal sides and four right angles.

• Side length = a
• Perimeter = 4a
• Area = a²
• Diagonal = a√2

Example: If the side of a square is 5 cm, its area is 5² = 25 sq cm, and its perimeter is 4 * 5 = 20 cm.

2. Rectangle

A rectangle is a quadrilateral with opposite sides equal and four right angles.

• Length = l, Breadth = b
• Perimeter = 2(l + b)
• Area = l × b
• Diagonal = √(l² + b²)

Example: A rectangle has a length of 10 cm and a breadth of 6 cm. Its area is 10 * 6 = 60 sq cm, and its perimeter is 2(10 + 6) = 32 cm.

3. Triangle

A triangle is a polygon with three sides and three angles.

a. Equilateral Triangle

All sides are equal (a), and all angles are 60°.

• Perimeter = 3a
• Area = (√3/4) * a²
• Height = (√3/2) * a

b. Isosceles Triangle

Two sides are equal (a), and the third side is b. The angles opposite the equal sides are equal.

• Area = (b/4) * √(4a² - b²)

c. Scalene Triangle

All sides are different (a, b, c).

• Area (using Heron's formula): Let s = (a + b + c)/2 (semi-perimeter). Area = √[s(s-a)(s-b)(s-c)]

d. Right-Angled Triangle

One angle is 90°.

• Area = (1/2) * base * height
• Hypotenuse² = perpendicular² + base² (Pythagorean theorem)

Example: An equilateral triangle has a side of 6 cm. Its area is (√3/4) * 6² = 9√3 sq cm.

4. Parallelogram

A quadrilateral with opposite sides parallel and equal.

• Area = base × height
• Area = ab sin(θ), where a and b are adjacent sides and θ is the angle between them.

5. Rhombus

A parallelogram with all sides equal.

• Area = (1/2) * d₁ * d₂, where d₁ and d₂ are the lengths of the diagonals.
• Area = a² sin(θ), where a is the side length and θ is any interior angle.

6. Trapezium (Trapezoid)

A quadrilateral with one pair of opposite sides parallel.

• Area = (1/2) * (sum of parallel sides) * height
• Area = (1/2) * (a + b) * h

7. Circle

A set of points equidistant from a central point.

• Radius = r
• Diameter = 2r
• Circumference = 2πr
• Area = πr²
• Area of a semicircle = (1/2)πr²
• Perimeter of a semicircle = πr + 2r
• Area of a sector = (θ/360°) * πr², where θ is the angle in degrees.
• Arc length of a sector = (θ/360°) * 2πr

Note: π (pi) is approximately 22/7 or 3.14159.

3D Mensuration (Solid Figures)

1. Cuboid

A rectangular solid with six rectangular faces.

• Length = l, Breadth = b, Height = h
• Surface Area = 2(lb + bh + hl)
• Volume = l × b × h
• Length of the diagonal = √(l² + b² + h²)

Example: A cuboid with dimensions 10 cm x 5 cm x 4 cm has a volume of 10 * 5 * 4 = 200 cubic cm and a surface area of 2(10*5 + 5*4 + 4*10) = 2(50 + 20 + 40) = 2(110) = 220 sq cm.

2. Cube

A cuboid with all edges equal (a).

• Edge = a
• Surface Area = 6a²
• Volume = a³
• Length of the diagonal = a√3

3. Cylinder

A solid with two parallel circular bases connected by a curved surface.

• Radius of base = r, Height = h
• Curved Surface Area (CSA) = 2πrh
• Total Surface Area (TSA) = 2πrh + 2πr² = 2πr(h + r)
• Volume = πr²h

Example: A cylinder with radius 7 cm and height 10 cm has a volume of π * 7² * 10 = (22/7) * 49 * 10 = 1540 cubic cm.

4. Cone

A solid with a circular base and a vertex.

• Radius of base = r, Height = h, Slant height = l
• Relationship: l² = r² + h²
• Curved Surface Area (CSA) = πrl
• Total Surface Area (TSA) = πrl + πr² = πr(l + r)
• Volume = (1/3)πr²h

5. Sphere

A perfectly round geometrical object in three-dimensional space.

• Radius = r
• Surface Area = 4πr²
• Volume = (4/3)πr³

Note: A hemisphere has half the surface area and volume of a sphere. Its TSA includes the base area.

6. Hemisphere

Half of a sphere.

• Radius = r
• Curved Surface Area = 2πr²
• Total Surface Area = 2πr² + πr² = 3πr²
• Volume = (2/3)πr³

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Geometry

Geometry is the study of shapes, sizes, positions, angles, and dimensions of things. It is fundamental to understanding mensuration and trigonometry.

Lines and Angles

• Line: Extends infinitely in both directions.
• Line Segment: A part of a line with two endpoints.
• Ray: A line with one endpoint, extending infinitely in one direction.
• Angle: Formed by two rays sharing a common endpoint (vertex). Measured in degrees or radians.
• Types of Angles:
  • Acute Angle: > 0° and < 90°
  • Right Angle: = 90°
  • Obtuse Angle: > 90° and < 180°
  • Straight Angle: = 180°
  • Reflex Angle: > 180° and < 360°
• Complementary Angles: Two angles whose sum is 90°.
• Supplementary Angles: Two angles whose sum is 180°.
• Adjacent Angles: Share a common vertex and a common side, but no common interior points.
• Vertically Opposite Angles: Formed when two lines intersect. They are equal.
• Parallel Lines: Lines in a plane that do not meet.
• Transversal: A line that intersects two or more lines.
  • Corresponding Angles are equal.
  • Alternate Interior Angles are equal.
  • Alternate Exterior Angles are equal.
  • Consecutive Interior Angles are supplementary (sum to 180°).

Triangles (Geometric Properties)

• The sum of the angles in any triangle is always 180°.
• The sum of the lengths of any two sides of a triangle is always greater than the length of the third side (Triangle Inequality Theorem).
• Types of Triangles based on angles:
  • Acute-angled Triangle: All angles < 90°.
  • Right-angled Triangle: One angle = 90°.
  • Obtuse-angled Triangle: One angle > 90°.
• Types of Triangles based on sides:
  • Equilateral Triangle: All sides equal, all angles 60°.
  • Isosceles Triangle: Two sides equal, angles opposite equal sides are equal.
  • Scalene Triangle: All sides different, all angles different.
• Congruent Triangles: Triangles are congruent if their corresponding sides and angles are equal. Conditions for congruence (SSS, SAS, ASA, AAS, RHS).
• Similar Triangles: Triangles are similar if their corresponding angles are equal, and the ratio of their corresponding sides is constant. Conditions for similarity (AAA, AA, SSS similarity, SAS similarity).
• Centroid: Intersection of medians. Divides each median in a 2:1 ratio.
• Incenter: Intersection of angle bisectors. Center of the inscribed circle.
• Circumcenter: Intersection of perpendicular bisectors of sides. Center of the circumscribed circle.
• Orthocenter: Intersection of altitudes.

Quadrilaterals

• The sum of the interior angles of any quadrilateral is 360°.
• Parallelogram: Opposite sides parallel and equal; opposite angles equal; diagonals bisect each other.
• Rectangle: A parallelogram with four right angles; diagonals are equal and bisect each other.
• Rhombus: A parallelogram with four equal sides; diagonals are perpendicular bisectors of each other and bisect the angles.
• Square: A rectangle with equal sides; a rhombus with right angles; diagonals are equal, perpendicular bisectors, and bisect angles.
• Trapezium: One pair of parallel sides.
• Kite: Two pairs of equal adjacent sides; one diagonal is the perpendicular bisector of the other; one diagonal bisects the angles at its endpoints.

Circles (Geometric Properties)

• Radius: Distance from the center to any point on the circle.
• Diameter: A line segment passing through the center, with endpoints on the circle (Diameter = 2 × Radius).
• Chord: A line segment whose endpoints lie on the circle.
• Secant: A line that intersects the circle at two points.
• Tangent: A line that touches the circle at exactly one point (point of contact). The radius drawn to the point of contact is perpendicular to the tangent.
• Arc: A portion of the circumference of a circle.
• Sector: The region bounded by two radii and the intercepted arc.
• Segment: The region bounded by a chord and the intercepted arc.
• Angles in a Circle:
  • The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
  • Angles in the same segment are equal.
  • The angle in a semicircle is a right angle (90°).
• Properties of Chords:
  • Equal chords subtend equal angles at the center.
  • A perpendicular from the center to a chord bisects the chord.
  • Chords equidistant from the center are equal.
• Power of a Point Theorem (Intersecting Chords, Secants, Tangents):
  • Intersecting Chords: If two chords AB and CD intersect at a point P inside a circle, then AP × PB = CP × PD.
  • Intersecting Secants: If two secants from an external point P intersect the circle at A, B and C, D respectively, then PA × PB = PC × PD.
  • Secant and Tangent: If a secant from an external point P intersects the circle at A, B and a tangent from P touches the circle at T, then PA × PB = PT².

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Trigonometry

Trigonometry is the branch of mathematics concerned with relationships between angles and sides of triangles. It is particularly useful for solving problems involving heights and distances.

Basic Concepts (Right-Angled Triangle)

Consider a right-angled triangle ABC, with the right angle at B. Let θ (theta) be one of the acute angles, say at vertex A.

• Hypotenuse (H): The side opposite the right angle (AC).
• Perpendicular (P) / Opposite: The side opposite the angle θ (BC).
• Base (B) / Adjacent: The side adjacent to the angle θ (AB).

The trigonometric ratios are defined as follows:

Ratio	Abbreviation	Definition
Sine	sin(θ)	Perpendicular / Hypotenuse (P/H)
Cosine	cos(θ)	Base / Hypotenuse (B/H)
Tangent	tan(θ)	Perpendicular / Base (P/B)
Cosecant	cosec(θ) or csc(θ)	Hypotenuse / Perpendicular (H/P) = 1/sin(θ)
Secant	sec(θ)	Hypotenuse / Base (H/B) = 1/cos(θ)
Cotangent	cot(θ)	Base / Perpendicular (B/P) = 1/tan(θ)

Trigonometric Identities

These are equations that are true for all values of the variables involved.

• Fundamental Identity: sin²(θ) + cos²(θ) = 1
• From this, we derive:
  • sin²(θ) = 1 - cos²(θ)
  • cos²(θ) = 1 - sin²(θ)
• Other Important Identities:
  • 1 + tan²(θ) = sec²(θ)
  • 1 + cot²(θ) = cosec²(θ)
• Relationship between ratios: tan(θ) = sin(θ) / cos(θ) ; cot(θ) = cos(θ) / sin(θ)

Values of Trigonometric Ratios for Standard Angles

Memorizing these values is crucial for solving problems quickly.

Angle (θ)	sin(θ)	cos(θ)	tan(θ)	cosec(θ)	sec(θ)	cot(θ)
0°	0	1	0	Undefined	1	Undefined
30°	1/2	√3/2	1/√3	2	2/√3	√3
45°	1/√2	1/√2	1	√2	√2	1
60°	√3/2	1/2	√3	2/√3	2	1/√3
90°	1	0	Undefined	1	Undefined	0

Trigonometric Ratios of Complementary Angles

If A and B are complementary angles (A + B = 90°), then:

• sin(A) = cos(B)
• cos(A) = sin(B)
• tan(A) = cot(B)
• cot(A) = tan(B)
• sec(A) = cosec(B)
• cosec(A) = sec(B)

Example: sin(30°) = cos(60°), tan(20°) = cot(70°).

Heights and Distances

This is a direct application of trigonometry, especially using right-angled triangles.

• Angle of Elevation: The angle formed between the horizontal line from the observer's eye and the line of sight to an object above the horizontal line.
• Angle of Depression: The angle formed between the horizontal line from the observer's eye and the line of sight to an object below the horizontal line.

Common Scenarios:

• To find the height of a tower/building: Use tan ratio. Height = Distance × tan(Angle of Elevation).
• To find the distance from the base of a tower: Use cot ratio. Distance = Height × cot(Angle of Elevation).
• Problems often involve angles like 30°, 45°, 60°.

Example: The angle of elevation of the top of a tower from a point 100 meters away from its base is 30°. Find the height of the tower.

Let height = h, distance = 100 m, angle = 30°.

tan(30°) = height / distance

1/√3 = h / 100

h = 100 / √3 = (100√3) / 3 meters.

Trigonometric Ratios for angles greater than 90°

These are determined using the unit circle and reference angles. The signs of the ratios depend on the quadrant the angle lies in.

• Quadrant I (0° - 90°): All ratios are positive.
• Quadrant II (90° - 180°): Sine and Cosecant are positive; others are negative.
• Quadrant III (180° - 270°): Tangent and Cotangent are positive; others are negative.
• Quadrant IV (270° - 360°): Cosine and Secant are positive; others are negative.

Rules for changing angles:

• If the angle is 90° ± θ or 270° ± θ, the ratio changes (sin → cos, cos → sin, tan → cot, cot → tan, sec → cosec, cosec → sec).
• If the angle is 180° ± θ or 360° ± θ, the ratio remains the same.
• The sign depends on the quadrant of the original angle.

Example: sin(150°) = sin(180° - 30°) = sin(30°) = 1/2. (150° is in Quadrant II, where sine is positive).

cos(120°) = cos(180° - 60°) = -cos(60°) = -1/2. (120° is in Quadrant II, where cosine is negative).

Sine Rule and Cosine Rule (For any triangle, not just right-angled)**

For a triangle ABC with sides a, b, c opposite to angles A, B, C respectively:

• Sine Rule: a/sin(A) = b/sin(B) = c/sin(C) = 2R (where R is the circumradius)
• Cosine Rule:
  • a² = b² + c² - 2bc cos(A)
  • b² = a² + c² - 2ac cos(B)
  • c² = a² + b² - 2ab cos(C)

These rules are useful for solving triangles where you don't have a right angle or when dealing with more complex geometric problems.