Mensuration of Plane and Solid Figures

Welcome to the comprehensive study of Mensuration! In this section, we will delve deep into the world of geometric shapes, focusing on calculating their areas, perimeters, volumes, and surface areas. Mensuration is a crucial topic in mathematics, especially for competitive exams like SSC CGL, as it tests your ability to visualize shapes and apply formulas accurately. We will cover both plane figures (2D shapes) and solid figures (3D shapes).

Part 1: Mensuration of Plane Figures (2D Shapes)

Plane figures are shapes that lie on a flat surface and have only two dimensions: length and breadth (or height). We will explore common plane figures and their properties.

1. Triangles

A triangle is a polygon with three sides and three angles. The sum of its interior angles is always 180 degrees.

1.1 Types of Triangles and Their Area Formulas

• Equilateral Triangle: All sides are equal (a). All angles are 60 degrees.
  • Area = (√3 / 4) * a²
  • Perimeter = 3a
  • Height = (√3 / 2) * a
• Isosceles Triangle: Two sides are equal. The angles opposite the equal sides are also equal.
  • Let the equal sides be 'a' and the base be 'b'.
  • Area = (1/2) * b * h , where h is the altitude to the base.
  • Using Pythagoras theorem on half of the base: h = √(a² - (b/2)²)
  • Area = (b/4) * √(4a² - b²)
  • Perimeter = 2a + b
• Scalene Triangle: All sides are of different lengths (a, b, c).
  • Area (using Heron's formula): Let s = semi-perimeter = (a+b+c)/2 . Area = √[s(s-a)(s-b)(s-c)]
  • Perimeter = a + b + c
• Right-angled Triangle: One angle is 90 degrees. Sides are perpendicular.
  • Let the base be 'b' and height be 'h' (the two sides forming the right angle). The third side is the hypotenuse 'c'.
  • Area = (1/2) * b * h
  • Perimeter = b + h + c
  • Pythagorean Theorem: a² + b² = c²
• Area using Trigonometry: Area = (1/2) * ab * sin(C) , where C is the angle between sides 'a' and 'b'.

2. Quadrilaterals

A quadrilateral is a polygon with four sides and four angles. The sum of its interior angles is 360 degrees.

2.1 Types of Quadrilaterals and Their Area Formulas

• Rectangle: Opposite sides are equal and parallel. All angles are 90 degrees.
  • Let length = l, breadth = b.
  • Area = l * b
  • Perimeter = 2(l + b)
  • Diagonal = √(l² + b²)
• Square: All sides are equal (a). All angles are 90 degrees.
  • Area = a²
  • Perimeter = 4a
  • Diagonal = a√2
  • Area can also be calculated using diagonal 'd': Area = d² / 2
• Parallelogram: Opposite sides are equal and parallel. Opposite angles are equal.
  • Let base = b, height = h.
  • Area = b * h
  • Perimeter = 2(a + b), where a and b are adjacent sides.
• Rhombus: All sides are equal. Opposite angles are equal. Diagonals bisect each other at right angles.
  • Let diagonals be d1 and d2.
  • Area = (1/2) * d1 * d2
  • Perimeter = 4a (where 'a' is the side length)
  • Side 'a' can be found using diagonals: a = √[(d1/2)² + (d2/2)²]
• Trapezium (or Trapezoid): One pair of opposite sides is parallel.
  • Let parallel sides be 'a' and 'b', and height be 'h' (perpendicular distance between parallel sides).
  • Area = (1/2) * (a + b) * h
  • Perimeter = sum of all four sides.
• Kite: Two pairs of adjacent sides are equal. One pair of opposite angles is equal. Diagonals are perpendicular.
  • Area = (1/2) * d1 * d2

3. Circles

A circle is a set of points equidistant from a central point. It has only one dimension: radius.

3.1 Circle Formulas

• Let radius = r, diameter = d (d = 2r), circumference = C.
• Area = πr²
• Circumference = 2πr or πd
• Area of a Semicircle = (1/2)πr²
• Circumference of a Semicircle = πr + 2r (arc length + diameter)
• Area of a Sector (with angle θ in degrees) = (θ/360) * πr²
• Length of an Arc (with angle θ in degrees) = (θ/360) * 2πr
• Area of a Segment = Area of Sector - Area of Triangle formed by the radii and the chord.

Value of π (Pi): Approximately 22/7 or 3.14159.

4. Polygons (General)

A polygon is a closed shape made of straight line segments. A regular polygon has all sides and all angles equal.

4.1 Formulas for Regular Polygons

• Let n = number of sides, a = length of each side.
• Sum of interior angles = (n-2) * 180°
• Each interior angle = [(n-2) * 180°] / n
• Sum of exterior angles = 360°
• Each exterior angle = 360° / n
• Area of a Regular Polygon = (1/4) * n * a² * cot(180°/n)
• Area of a Regular Polygon (using apothem 'ap') = (1/2) * Perimeter * ap

Part 2: Mensuration of Solid Figures (3D Shapes)

Solid figures are three-dimensional objects that have length, breadth, and height. We will focus on their volume (space occupied) and surface area (total area of all faces).

1. Cuboid

A cuboid is a rectangular prism. It has 6 rectangular faces, 12 edges, and 8 vertices.

• Let length = l, breadth = b, height = h.
• Volume (V) = l * b * h
• Total Surface Area (TSA) = 2(lb + bh + hl)
• Lateral Surface Area (LSA) = 2(lh + bh) = 2(l + b)h (Area of the four walls)
• Length of Diagonal = √(l² + b² + h²)

2. Cube

A cube has six equal square faces.

• Let side length = a.
• Volume (V) = a³
• Total Surface Area (TSA) = 6a²
• Lateral Surface Area (LSA) = 4a²
• Length of Diagonal = a√3

3. Cylinder

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

• Let radius of base = r, height = h.
• Volume (V) = Area of base * height = πr²h
• Curved Surface Area (CSA) = Circumference of base * height = 2πrh
• Total Surface Area (TSA) = CSA + Area of two bases = 2πrh + 2πr² = 2πr(h + r)

4. Cone

A cone is a solid figure with a circular base and a vertex. The distance from the vertex to any point on the circumference of the base is the slant height.

• Let radius of base = r, height = h, slant height = l.
• Relationship between r, h, and l: l² = r² + h² (Pythagoras theorem)
• Volume (V) = (1/3) * Area of base * height = (1/3)πr²h
• Curved Surface Area (CSA) = πrl
• Total Surface Area (TSA) = CSA + Area of base = πrl + πr² = πr(l + r)

5. Sphere

A sphere is a perfectly round geometrical object in three-dimensional space.

• Let radius = r.
• Volume (V) = (4/3)πr³
• Surface Area (SA) = 4πr²

6. Hemisphere

A hemisphere is half of a sphere.

• Let radius = r.
• Volume (V) = (1/2) * (4/3)πr³ = (2/3)πr³
• Curved Surface Area (CSA) = (1/2) * 4πr² = 2πr²
• Total Surface Area (TSA) = CSA + Area of the circular base = 2πr² + πr² = 3πr²

7. Frustum of a Cone

A frustum is the part of a cone left after cutting off the top part with a plane parallel to the base.

• Let radii of the two bases be R and r (R > r), height = h, slant height = l.
• Relationship between R, r, h, and l: l² = h² + (R-r)²
• Volume (V) = (1/3)πh(R² + Rr + r²)
• Curved Surface Area (CSA) = πl(R + r)
• Total Surface Area (TSA) = CSA + Area of two bases = πl(R + r) + π(R² + r²)

8. Prism and Pyramid

These are general categories of solids.

• Prism: A solid with two identical and parallel bases, and rectangular sides connecting corresponding edges of the bases.
  • Volume = Area of Base * Height
  • Lateral Surface Area = Perimeter of Base * Height
• Pyramid: A solid with a polygonal base and triangular faces that meet at a point (apex).
  • Volume = (1/3) * Area of Base * Height
  • Lateral Surface Area = Sum of the areas of all triangular faces.

Part 3: Combined and Composite Shapes

Often, exam questions involve shapes made up of combinations of basic figures. The key is to break down the composite shape into its constituent parts and apply the appropriate formulas.

1. Strategy for Composite Shapes

1. Identify the constituent shapes: Recognize which basic 2D or 3D shapes make up the composite figure (e.g., a cylinder with hemispheres at both ends, a cube with a pyramid on top).
2. Determine what needs to be calculated: Are you finding the total volume, total surface area, or a specific area/volume?
3. Calculate individual components: Find the volume and/or surface area of each basic shape involved.
4. Combine results:
   • For Volume: Add the volumes of all parts. If one shape is removed from another (e.g., a conical cavity in a cylinder), subtract the volume of the removed part.
   • For Surface Area: This is trickier. Add the surface areas of the exposed parts. Crucially, any surface where two shapes are joined together is *not* part of the total surface area. For example, in a cylinder with hemispheres on top, the circular bases of the cylinder are covered by the hemispheres, so they are excluded from the TSA calculation. You would add the curved surface area of the cylinder and the curved surface area of the two hemispheres.

2. Example: Capsule Shape

A capsule is formed by a cylinder with a hemisphere attached to each of its circular ends.

• Let the radius of the cylinder and hemispheres be 'r', and the height of the cylindrical part be 'h'.
• Total Volume = Volume of Cylinder + Volume of 2 Hemispheres (which is equal to the volume of 1 sphere)
  • V = πr²h + (4/3)πr³
• Total Surface Area = Curved Surface Area of Cylinder + Surface Area of 2 Hemispheres (which is equal to the surface area of 1 sphere)
  • TSA = 2πrh + 4πr²

3. Example: Ice Cream Cone

An ice cream cone consists of a cone topped with a hemisphere.

• Let the radius of the cone's base and the hemisphere be 'r', and the height of the cone be 'h'. The slant height of the cone is 'l'.
• Total Volume = Volume of Cone + Volume of Hemisphere
  • V = (1/3)πr²h + (2/3)πr³
• Total Surface Area = Curved Surface Area of Cone + Curved Surface Area of Hemisphere
  • TSA = πrl + 2πr²

  Note: The circular base of the cone and the hemisphere are joined, so they are not included in the surface area.

Part 4: Key Concepts and Problem-Solving Tips

1. Units and Conversions

Always pay close attention to the units. If dimensions are given in different units (e.g., meters and centimeters), convert them to a consistent unit before calculation. The final answer's unit will depend on the input units (e.g., cm² for area, cm³ for volume).

2. Visualization

Try to visualize the shapes. Sketching the figure, especially for composite shapes, can greatly help in identifying the relevant dimensions and surfaces.

3. Formulas are Key

Memorize the essential formulas for basic shapes. Understanding the derivation or relationship between formulas (like cone vs. cylinder volume) can also aid recall.

4. Practice with Different Problems

Solve a variety of problems ranging from simple area/volume calculations to complex composite shape problems. This builds speed and accuracy.

5. Check for Special Cases

Recognize special cases like squares (special rectangles), cubes (special cuboids), and equilateral triangles, as they often simplify calculations.

Mastering mensuration requires a combination of understanding geometric principles, accurate formula application, and careful calculation. By systematically approaching each problem and practicing regularly, you will build confidence and proficiency in this vital area of mathematics.