Mensuration of Plane Figures, Mensuration of Solid Figures, Area and Volume Calculations

Welcome to the in-depth study of Mensuration, a crucial topic in Quantitative Aptitude. Mensuration deals with the measurement of geometrical figures. In this section, we will focus on calculating the areas and perimeters of plane figures and the surface areas and volumes of solid figures. Mastering these concepts will equip you to solve a variety of problems that often appear in competitive exams.

I. Mensuration of Plane Figures

Plane figures are two-dimensional shapes that lie on a flat surface. They have length and breadth but no depth. We will cover common plane figures, their properties, and the formulas for their area and perimeter. The perimeter is the total length of the boundary of a plane figure, while the area is the space enclosed within its boundary.

A. Triangles

A triangle is a polygon with three sides and three angles. The sum of the angles in any triangle is always 180 degrees.

1. Types of Triangles and Formulas

We encounter various types of triangles, each with specific properties and area calculation methods:

• Equilateral Triangle: All three sides are equal (a = b = c). All angles are 60 degrees.
  • Perimeter (P) = 3 * side (s)
  • Area (A) = (√3 / 4) * s²
• 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'.
  • Height (h) = √(a² - (b/2)²)
  • Perimeter (P) = 2a + b
  • Area (A) = (1/2) * base * height = (1/2) * b * h
• Scalene Triangle: All three sides are of different lengths.
  • Perimeter (P) = a + b + c
  • Area (A) = Using Heron's Formula:
    1. Calculate the semi-perimeter (s): s = (a + b + c) / 2
    2. Area (A) = √[s(s-a)(s-b)(s-c)]
• Right-Angled Triangle: One angle is 90 degrees. The side opposite the right angle is the hypotenuse.
  • Let the base be 'b' and height be 'h'. The hypotenuse is 'hyp'.
  • Perimeter (P) = b + h + hyp
  • Area (A) = (1/2) * base * height = (1/2) * b * h
  • By Pythagorean theorem: hyp² = b² + h²
• Acute-Angled Triangle: All angles are less than 90 degrees.
• Obtuse-Angled Triangle: One angle is greater than 90 degrees.

B. Quadrilaterals

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

1. Types of Quadrilaterals and Formulas

• Square: All four sides are equal (s), and all four angles are 90 degrees.
  • Perimeter (P) = 4 * s
  • Area (A) = s²
  • Diagonal (d) = s * √2
  • Area (A) = (1/2) * d² (using diagonal)
• Rectangle: Opposite sides are equal (length 'l' and breadth 'b'), and all four angles are 90 degrees.
  • Perimeter (P) = 2 * (l + b)
  • Area (A) = l * b
  • Diagonal (d) = √(l² + b²)
• Parallelogram: Opposite sides are parallel and equal. Opposite angles are equal.
  • Let sides be 'a' and 'b', and the angle between them be θ.
  • Perimeter (P) = 2 * (a + b)
  • Area (A) = base * height (h)
  • Area (A) = a * b * sin(θ)
• Rhombus: All four sides are equal (s). Opposite sides are parallel. Diagonals bisect each other at right angles.
  • Perimeter (P) = 4 * s
  • Let diagonals be d1 and d2.
  • Area (A) = (1/2) * d1 * d2
  • Also, s² = (d1/2)² + (d2/2)²
• Trapezium (Trapezoid): Exactly one pair of opposite sides is parallel.
  • Let parallel sides be 'a' and 'b', and the height be 'h'.
  • Area (A) = (1/2) * (sum of parallel sides) * height = (1/2) * (a + b) * h
• Kite: Two pairs of adjacent sides are equal. Diagonals are perpendicular, and one diagonal bisects the other.
  • Area (A) = (1/2) * d1 * d2

C. Circles

A circle is a set of points equidistant from a central point. The distance from the center to any point on the circle is the radius (r).

• Diameter (d) = 2 * r
• Circumference (C) = 2 * π * r or π * d (where π ≈ 22/7 or 3.14)
• Area (A) = π * r²
• Area of a Semicircle = (1/2) * π * r²
• Area of a Sector (with angle θ in degrees) = (θ / 360) * π * r²
• Length of an Arc (with angle θ in degrees) = (θ / 360) * 2 * π * r

II. Mensuration of Solid Figures

Solid figures are three-dimensional shapes that have length, breadth, and height. They occupy space and have volume and surface area. We will discuss different solid shapes, their formulas for surface area and volume.

A. Cuboid

A cuboid is a rectangular solid figure with six rectangular faces. It has length (l), breadth (b), and height (h).

• Volume (V) = l * b * h
• Total Surface Area (TSA) = 2 * (lb + bh + hl)
• Lateral Surface Area (LSA) = 2 * (l + b) * h (Area of the four walls)
• Length of the Diagonal (d) = √(l² + b² + h²)

B. Cube

A cube is a special type of cuboid where all six faces are squares, and all edges are equal (side 's').

• Volume (V) = s³
• Total Surface Area (TSA) = 6 * s²
• Lateral Surface Area (LSA) = 4 * s²
• Length of the Diagonal (d) = s * √3

C. Cylinder

A cylinder is a solid figure with two identical circular bases joined by a curved surface. It has radius (r) and height (h).

• Volume (V) = Area of base * height = π * r² * h
• Curved Surface Area (CSA) = Circumference of base * height = 2 * π * r * h
• Total Surface Area (TSA) = CSA + Area of two bases = 2πrh + 2πr² = 2πr(h + r)

D. Cone

A cone is a solid figure that tapers from a circular base to a point called the apex. It has radius (r) of the base, height (h) (perpendicular distance from apex to base), and slant height (l).

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

E. Sphere

A sphere is a perfectly round geometrical object in three-dimensional space. It is defined by its radius (r).

• Volume (V) = (4/3) * π * r³
• Surface Area (SA) = 4 * π * r²

F. Hemisphere

A hemisphere is half of a sphere.

• Volume (V) = (1/2) * Volume of sphere = (2/3) * π * r³
• Curved Surface Area (CSA) = (1/2) * Surface Area of sphere = 2 * π * r²
• Total Surface Area (TSA) = CSA + Area of the circular base = 2πr² + πr² = 3πr²

G. Frustum of a Cone

A frustum is the part of a cone that remains when the top portion is cut off by a plane parallel to the base.

• Let the radii of the two circular bases be R and r (R > r), and the height be h.
• Slant height (l) = √[h² + (R - r)²]
• Volume (V) = (1/3) * π * h * (R² + Rr + r²)
• Curved Surface Area (CSA) = π * (R + r) * l
• Total Surface Area (TSA) = CSA + Area of top base + Area of bottom base = π(R + r)l + πR² + πr² = π[(R + r)l + R² + r²]

III. Area and Volume Calculations - Practice and Application

Understanding the formulas is the first step. The real challenge lies in applying them correctly to solve problems. Many questions involve combining different shapes or finding dimensions based on given areas or volumes.

A. Composite Figures

Some problems involve figures made up of two or more basic geometric shapes. To find the area or volume of such figures, you need to identify the individual shapes, calculate their respective areas or volumes, and then add or subtract them as required.

• Example: A toy is in the shape of a cone mounted on a hemisphere. The radius of the hemisphere is 3 cm, and the total height of the toy is 8 cm. Find the volume of the toy.
  • Radius of hemisphere (r) = 3 cm.
  • Radius of cone = Radius of hemisphere = 3 cm.
  • Height of hemisphere = radius = 3 cm.
  • Height of cone (h_cone) = Total height - Height of hemisphere = 8 cm - 3 cm = 5 cm.
  • Volume of Hemisphere = (2/3) * π * r³ = (2/3) * π * (3)³ = (2/3) * π * 27 = 18π cubic cm.
  • Volume of Cone = (1/3) * π * r² * h_cone = (1/3) * π * (3)² * 5 = (1/3) * π * 9 * 5 = 15π cubic cm.
  • Total Volume of Toy = Volume of Hemisphere + Volume of Cone = 18π + 15π = 33π cubic cm.

B. Problems Involving Conversions

Sometimes, a solid object is melted and recast into another solid shape. In such cases, the volume of the material remains constant. This principle is often used to find unknown dimensions.

• Example: A solid metallic sphere of radius 6 cm is melted and recast into a cone with a base radius of 3 cm. Find the height of the cone.
  • Volume of Sphere = (4/3) * π * (6)³ = (4/3) * π * 216 = 288π cubic cm.
  • Volume of Cone = (1/3) * π * r² * h = (1/3) * π * (3)² * h = (1/3) * π * 9 * h = 3πh cubic cm.
  • Since volume is conserved: Volume of Sphere = Volume of Cone
  • 288π = 3πh
  • h = 288π / 3π = 96 cm.

C. Surface Area vs. Volume Problems

Be careful to distinguish between surface area and volume. Surface area is a measure of the total area that the surface of an object occupies, while volume is the amount of space it occupies.

• Example: A cuboid has dimensions 10 cm x 8 cm x 6 cm. Find the ratio of its total surface area to its volume.
  • TSA = 2 * (lb + bh + hl) = 2 * (10*8 + 8*6 + 6*10) = 2 * (80 + 48 + 60) = 2 * 188 = 376 sq cm.
  • Volume (V) = l * b * h = 10 * 8 * 6 = 480 cubic cm.
  • Ratio (TSA : V) = 376 : 480. Simplify by dividing by common factors. Both are divisible by 8: 47 : 60.

D. Practical Applications

Mensuration concepts are widely used in real life. For example:

• Calculating the amount of paint needed for walls (LSA of walls).
• Determining the capacity of containers like tanks or cans (Volume).
• Finding the amount of fabric needed for a cylindrical tent (CSA).
• Estimating the amount of material needed to construct a room or a building (Volume and Surface Area).