Mensuration is the part of geometry concerned with ascertaining lengths, areas, and volumes.
Perimeter is the distance around a closed figure. Perimeter or length is measured in linear units. For example, units for perimeter (or length) are m or cm or mm.
Area is the part of plane or region occupied by the closed figure. Area is measured in square units. For example, units for area are m2 or cm2 or mm2.
1. Triangle
Sides: a, b, c
Perimeter = a + b + c
Area Δ = 1/2 × Base × Height
$$ \Delta = \sqrt{s(s - a)(s - b)(s - c)} $$
Example: The base of a triangular field is three times its corresponding altitude. If the cost of ploughing the field at the rate of Rs.15 per square metre is Rs.20250, find the base and the corresponding altitude of the field.
2. Rectangle
Length (l), Breadth (b)
Perimeter = 2(Length + Breadth) = 2(l + b)
Area = Length × Breadth = l × b
Example: Length and breadth of a rectangular field are 23.7 m and 14.5 m respectively. Find perimeter and area of the field.
3. Square
Side: a
Perimeter = 4a
Area = a2
Example: Find the area of square whose perimeter is 80 m.
4. Parallelogram
Perimeter = 2(a + b)
Area = Base × Corresponding height = bh
Example: Find the area of a parallelogram of base 12 cm and corresponding altitude 8 cm.
5. Trapezium
Perimeter = a + b + c + d
Area = (half the sum of parallel sides) × height = ½(a + b)h
Example: Length of the two parallel sides of a trapezium are 20 cm and 12 cm and the distance between them is 5 cm. Find the area of the trapezium.
6. Rhombus
Perimeter = 4a
Area = ½ × d1 × d2
Example: Find the area of a rhombus whose diagonals are of lengths 16 cm and 12 cm.
7. Circle
Radius: r
Circumference = 2πr
Area = πr2
Example: The radii of two circles are 18 cm and 10 cm. Find the radius of the circle whose circumference is equal to the sum of the circumferences of these two circles.
8. Sector of Circle
A part of a circular region enclosed between two radii of the corresponding circle is called a sector of the circle.
Perimeter of the sector
The perimeter of the sector OAPB is equal to OA + OB + length of arc APB.
$$ \text{Perimeter} = \frac{2 \pi r}{360} \times \theta $$
$$ \text{Area} = \frac{\pi r^2}{360} \times \theta $$
Example: Find the perimeter and area of the sector of a circle of radius 9 cm with central angle 35°.
Example: Find the perimeter and area of the sector of a circle of radius 6 cm and length of the arc of the sector as 22 cm.
Heron's Formula
If the base and corresponding altitude of a triangle are known, you can calculate the area of triangle. However, sometimes you are not given the altitude (height) corresponding to the given base of a triangle. Instead, you are given the three sides of the triangle.
Example: Find the area of the triangle ABC, whose sides AB, BC and CA are respectively 5 cm, 6 cm and 7 cm.
Example: The sides of a triangular field are 165 m, 154 m and 143 m. Find the area of the field.
Example: Find the area of a trapezium whose parallel sides are of lengths 11 cm and 25 cm and whose non-parallel sides are of lengths 15 cm and 13 cm.
Rectangular and Circular Paths
Example: A rectangular park of length 30 m and breadth 24 m is surrounded by a 4 m wide path. Find the area of the path.
Example: There are two rectangular paths in the middle of a park. Find the cost of paving the paths with concrete at the rate of Rs. 15 per m2. It is given that AB = CD = 50 m, AD = BC = 40 m and EF = PQ = 2.5 m.
Example: There is a circular path of width 2 m along the boundary and inside a circular park of radius 16 m. Find the cost of paving the path with bricks at the rate of Rs. 24 per m2. (Use π = 3.14)
Combination of Figures
Example: In a round table cover, a design is made leaving an equilateral triangle ABC in the middle. If the radius of the cover is 3.5 cm, find the cost of making the design at the rate of Rs. 0.50 per cm2.
Example: On a square shaped handkerchief, nine circular designs, each of radius 7 cm, are made. Find the area of the remaining portion of the handkerchief.