Mensuration

The learning objectives of this chapter in mathematics typically include understanding the fundamental concepts of mensuration which include perimeter, area, volume, etc. and the relationships among them.

Mensuration

Mensuration is a branch of mathematics that deals with the measurement of geometric quantities such as length, perimeter, area, volume and angles. It has numerous applications in both mathematics and everyday life.

Perimeter

The perimeter of a shape is the sum of all the distances along its boundary. In mathematical terms, it is denoted by the symbol 'P'.

Perimeter = Sum of all sides

The units of measurement used for expressing perimeter can include centimetres (cm), metres (m) and other length units.

Area

Area is the quantity of space contained within a closed shape. In mathematical terms, it is denoted by the symbol 'A'.

The units of measurement used for expressing area can include square centimetres (cm²) and square metres (m²).

Example:  What is the area of the square PQRS if the perimeter of a square PQRS is twice the perimeter of △XYZ?

a) 42.25 cm2
b) 42.75 cm2
c) 52.25 cm2
d) 52.75 cm2

Answer: a) 42.25 cm2 

Explanation: Perimeter of △XYZ
= XY + YZ + YZ
= 5.5 + 3.55 + 3.95
= 13 cm

Perimeter of a square PQRS is twice the perimeter of △XYZ.

Perimeter of a square PQRS = 2 × Perimeter of △XYZ
⇒ 4 × Side = 2 × 13 cm
⇒ 4 × Side = 26 cm                       
⇒ Side = 26 cm/4
⇒ Side = 6.5 cm

Area of a square = Side × Side 

= 6.5 cm × 6.5 cm 
= 42.25 cm2

Steps for calculating the area of a figure using squared paper

1. Begin by outlining the shape's perimeter on a square grid paper.
2. Disregard any area that is less than half a square in size.
3. If more than half a square lies within the shape, count it as a whole square.
4. When exactly half of a square is within the shape, consider its area as ½ square unit.
5. Finally, sum up these counted squares and halves to determine the area of the given enclosed shape.

The figure shows the area of a figure using squared paper.

Example: What is the difference in the area of the given shapes?

a) 1 cm²
b) 2 cm²
c) 3 cm²
d) 4 cm²

Answer: d) 4 cm²

Explanation: Area of each square = 1 cm × 1 cm = 1 cm²

For First Figure:

Total area = 6 cm²  + 8 cm²  = 14 cm² 
For Second Figure:

Total area = 9 cm² + 9 cm² = 18 cm²  

Difference in their area = Area of second figure − Area of first figure
                                   = 18 cm²   − 14 cm² 
                                    = 4 cm²