Class 6 Mathematics · Chapter 6 NotesPerimeter and Area

This chapter is about measuring shapes in two ways. The first is the distance around a shape, called the perimeter. The second is the amount of flat surface a shape covers, called the area. Both ideas are used in everyday life, such as when we fence a field, frame a picture, tile a floor or paint a wall. Students learn to measure lengths in centimetres and metres, and to find the perimeter and area of rectangles, squares and other simple figures. They also learn that perimeter is measured in ordinary length units, while area is measured in square units. The chapter encourages careful counting, measuring and comparing rather than memorising answers.

• Perimeterdistance around a closed figure
• Areaamount of surface covered by a closed figure
• Perimeter is measured in cm, m, etc.
• Area is measured in square units such as cm² and m²

The perimeter of a closed figure is the total length of its boundary. Imagine walking once around a park along its edge; the distance you cover is the perimeter. To find it, we add the lengths of all the sides of the figure. For shapes with straight sides, this means adding each side length carefully. The perimeter does not depend on the shape being regular; even an irregular closed figure has a perimeter, found by adding all its side lengths. Perimeter is always measured in units of length, such as centimetres (cm) or metres (m). If the sides are given in different units, we first change them to the same unit before adding.

• Perimeter = sum of the lengths of all sides
• It is measured in cm, m or other length units
• All side lengths must be in the same unit before adding

A rectangle has two pairs of equal opposite sides. If its length is l and its breadth is b, then its perimeter is l + b + l + b, which is 2 × (l + b). For example, a rectangle of length 8 cm and breadth 5 cm has perimeter 2 × (8 + 5) = 26 cm. A square is a special rectangle in which all four sides are equal. If each side is s, then the perimeter is 4 × s. So a square of side 6 cm has perimeter 4 × 6 = 24 cm. These formulas save time because we do not need to add every side separately each time. They work only when the measurements are in the same unit.

• Perimeter of a rectangle = 2 × (length + breadth)
• Perimeter of a square = 4 × side
• Examplerectangle 8 cm by 5 cm has perimeter 26 cm
• Examplesquare of side 6 cm has perimeter 24 cm

Not all shapes are rectangles or squares. Triangles, pentagons, hexagons and other polygons have perimeters found by adding the lengths of their sides. For a triangle, add the three side lengths. For a regular polygon, where all sides are equal, multiply the side length by the number of sides. For example, a regular pentagon of side 4 cm has perimeter 5 × 4 = 20 cm. Some shapes may have curved boundaries. In such cases, a simple formula may not work, and we may need to measure the boundary using a string or a thread and then measure the string with a ruler. This shows that perimeter is a general idea that applies to many kinds of closed figures.

• Perimeter of a triangle = sum of its three sides
• Perimeter of a regular polygon = number of sides × side length
• Curved boundaries can be measured using a string

Area is the amount of flat surface a closed figure covers. We measure area by counting how many unit squares fit inside the figure. A unit square is a square whose side is 1 unit long. If a shape covers 12 unit squares, its area is 12 square units. Area is measured in square units such as square centimetres (cm²) or square metres (m²). Two shapes can have the same area but look different, because area depends only on the surface covered, not on the shape's outline. Comparing areas helps us understand that a long thin shape and a short wide shape may cover the same amount of surface. Counting squares is a good way to build the idea before learning formulas.

• Area = number of unit squares needed to cover a shape
• Measured in square unitscm², m², etc.
• Different shapes can have the same area

For a rectangle, the area is found by multiplying its length by its breadth. If length = l and breadth = b, then area = l × b. For example, a rectangle 7 cm long and 4 cm wide has area 7 × 4 = 28 cm². This works because the rectangle can be divided into rows and columns of unit squares. A square is a rectangle with equal sides, so its area is side × side, or s². A square of side 5 cm has area 5 × 5 = 25 cm². When we multiply two lengths in centimetres, the result is in square centimetres. It is important to use the same unit for both length and breadth before multiplying.

• Area of a rectangle = length × breadth
• Area of a square = side × side
• Examplerectangle 7 cm by 4 cm has area 28 cm²
• Examplesquare of side 5 cm has area 25 cm²

Many shapes can be divided into rectangles and squares, and their areas can be found by adding the areas of these parts. For example, an L-shaped figure can be split into two rectangles. We find the area of each rectangle and add them. Sometimes a shape can be enclosed in a rectangle, and the unwanted parts subtracted. Counting unit squares on a grid is another useful method for irregular shapes. This approach shows that area is additive: if a shape is made of non-overlapping parts, its total area is the sum of the areas of the parts. These ideas help students estimate areas of everyday shapes such as leaves, tiles and rooms.

• Split a shape into rectangles and squares, then add their areas
• Subtract areas of unwanted parts if a shape is inside a rectangle
• Count unit squares on a grid for irregular shapes
• Area of a combined shape = sum of areas of its non-overlapping parts