How Many Squares? Class 5 Notes Maths Chapter 3

Sol: Yes, on comparing my hand with my father’s palm his palm is bigger than mine and also my father’s hand is longer than my hand.

Counting Grids

• To find the area and perimeter of any object we use the counting grids method. 
• The area of a plane figure is the number of squares required to completely cover it.
• The perimeter of a plane figure can be measured by counting the sides/edges of the squares that form the boundary.

Finding the Area

• Let's find the area of the grid given below. 
• We can simply count the total number of small squares present in it, because the area is nothing but space occupied by the shape of the object.
• Therefore, the area of the grid = 16 square unit

Example: Count the area of stamps C and E, assuming that one box on the grid paper makes up 1 square cm.

Sol: To find the area, count the number of squares inside the stamps. 
Area of Stamp C: There are six boxes that make up the stamp. Therefore, the area is 6 square cm.
Area of Stamp E: Stamp E is made up of four boxes. The area of this stamp is 4 square cm.

Relationship Between Area and Perimeter

• Consider two rectangles A and B having the same perimeter as 18 units.
  
  Rectangle A
  
  Rectangle B
• Observe that both of them have different areas (Area of Rectangle A = 18 square units whereas Area of Rectangle B is 20 square units)
  
  Rectangle A
  
  Rectangle B

Hence we conclude that:

• The two shapes having the same perimeter can have different areas. 
• Similarly, two shapes having the same areas can have different perimeters.

Area of Triangles

• A triangle is a 3-sided polygon, which is formed when three straight lines meet. 
• All triangles have three sides and three corners (angles). 
• Examples of triangles in real life: pyramids, sailing boats, sandwich and pizza slices etc.

Finding the Area of a Triangle

• Look at the green triangle below. Can you guess how can we find the area of the triangle inside the rectangle?

• The area of the rectangle = 2 square cm
• The triangle is exactly half of the rectangle, dividing the rectangle into two equal triangles.
• Therefore, the area of the triangle is half of the area of the rectangle, i.e., 1 square cm.

Sol: 

(i) The blue triangle: It divides the rectangle into two equal parts. Therefore, the area of the triangle is half of that of the rectangle. 
Area of rectangle = 20 square cm
Area of blue triangle = 10 square cm

(ii) The red triangle: The red triangle divides two different rectangles into half as shown in the picture: 

The area of the triangle = half of the area of the yellow rectangle + half of the area of the green rectangle
= half of 12 square cm + half of 8 square cm
=6 + 4 square cm
= 10 square cm

Area of Irregular Shapes

• Irregular shapes are those that don't have equal sides or angles.
• To find the area of irregular shapes, we can break them into smaller familiar shapes.
• These smaller shapes can be polygons like squares, rectangles, triangles, etc.
• For example, if we have an irregular shape made up of polygons P, Q, R, S, T, and U, we find the area of each polygon and add them up to get the total area of the irregular shape i.e. area of the given irregular shape = area of T + area of P + area of Q + area of R + area of S + area of U   
  
• Another example: If we have a shape made up of polygons A, B, C, D, E, F, and G, we find the area of each and add them to find the total area i.e. area of the given irregular shape = area of A + area of B + area of C + area of D + area of E + area of F + area of G  
  
• Therefore, for irregular shapes, we use methods like breaking them into known shapes.
• We might count unit squares within the shape or decompose them into polygons for accurate area calculation.

Solved Example