Measure the boundary of a shape and the surface it covers.
The perimeter of a closed plane figure is the distance travelled once around its boundary. For a polygon, add the lengths of all its sides. Perimeter is a length, so units are centimetres, metres or another unit of length. A drawing may be irregular, but the same principle applies: follow the entire boundary without skipping or counting an edge twice. When a side length is missing, use the information about equal sides or the total perimeter to find it. Perimeter answers “how far around?”, not “how much surface?”
For a rectangle with length l and breadth b, opposite sides are equal, so P = l + b + l + b = 2l + 2b = 2(l+b). For a square with side s, P = 4s. These formulas are shortcuts derived from adding all the sides. Example: a rectangle of length 8 cm and breadth 3 cm has P = 2(8+3) = 22 cm. Example: a square with side 5 m has P = 4×5 = 20 m. Always keep units consistent before adding.
Area measures the amount of surface inside a closed boundary. A square unit, such as a square centimetre, is a square with each side 1 cm. Counting equal square units is a way to understand area. Area is expressed in square units, not ordinary length units. A shape may have the same perimeter as another shape but a different area, so the two measures answer different questions. When estimating area on a grid, count full squares and combine partial squares carefully rather than treating the boundary as a line-length measurement.
A rectangle can be tiled by equal unit squares arranged in rows and columns. If there are l units in each row and b rows, the total number of unit squares is l×b, giving area A = l×b for a rectangle. For a square of side s, A = s×s = s². Example: a rectangle 7 cm by 4 cm has area 7×4 = 28 cm². Example: a square of side 6 cm has area 6×6 = 36 cm². The units multiply too: cm×cm becomes cm².
A triangle can be understood by comparing it with a rectangle or parallelogram built using the same base and height. The area relationship used in the chapter is half the product of base and corresponding height: A = ½ × base × height. The height is the perpendicular distance to the base, not necessarily one of the sloping sides. Example: if base = 8 cm and height = 5 cm, A = ½×8×5 = 20 cm². Write the units as square units and identify the perpendicular height before substituting values.
Example 1: A rectangle of 9 m by 2 m has perimeter 2(9+2)=22 m and area 9×2=18 m². The numerical answers differ because the measures describe different things. Example 2: A triangle with base 10 cm and height 6 cm has area ½×10×6=30 cm². Example 3: If a square has perimeter 28 cm, its side is 28÷4=7 cm and its area is 7×7=49 cm². Common mistakes include writing area in cm, perimeter in cm², using the sloping side as triangle height, or multiplying length and breadth when perimeter is requested.
| Idea | How to use it | Check |
|---|---|---|
| Identify the rule | Describe what changes and what remains fixed. | Test more than one example. |
| Choose a representation | Use a diagram, number line, table or equation. | Labels and units are clear. |
| Explain the answer | Show the operation and each step. | The result fits the question. |
Example A. Read the question and name the relevant mathematical idea. Write the information given before choosing an operation. This prevents a familiar-looking number from being used in the wrong way.
Example B. Show the calculation in small steps. For example, when a calculation involves a product and a sum, compute the product first if the expression requires it, then add. Keep intermediate values visible so an arithmetic slip can be found.
Example C. Check the result using a second representation where possible: estimate it, draw a model, use a number line, or substitute it back into the original relationship. A check is especially useful when a question involves units, scale, direction or a geometric condition.
The distance around a closed figure.
Square units such as cm² or m².
Length multiplied by breadth.
The perpendicular height to the chosen base.
Perimeter and Area builds mathematical understanding through careful observation and clear representations. Identify the key quantities, shapes or relationships before choosing a method. Use diagrams, tables, number lines or equations to make thinking visible. Show calculations in order and keep labels and units consistent. Check whether the result satisfies the definition or rule in the question. Explain the result in a sentence so another learner can follow the reasoning.