Perimeter & Area
The distance around a shape and the space inside it, two different questions.
Two different measurements
Perimeter is the total distance around the edge of a shape. You measure it in length units, cm, m, km.
Area is the amount of surface inside a shape. You measure it in square units, cm², m².
The clearest way to hold them apart: perimeter is the fence, area is the grass.
The formulas
Rectangle
Perimeter = 2 × (length + width)Area = length × width
Square: a rectangle with equal sides.
Perimeter = 4 × sideArea = side × side
Triangle
Perimeter = add the three sidesArea = ½ × base × height
The height must be the perpendicular height, straight up from the base, not along a slanted side.
Why area is measured in squares
Area is literally a count of unit squares that fit inside a shape.
A rectangle 8 cm by 5 cm holds 5 rows of 8 squares, which is why the formula is length × width. The multiplication is a shortcut for counting.
That is also why the unit is cm²: you multiplied centimetres by centimetres.
Understanding this means you never have to guess whether to add or multiply. You just ask whether you are walking around the edge or filling the inside.
A worked rectangle
Question. A garden is 12 m long and 7 m wide. Find its perimeter and area.
Perimeter: walk all the way round.
2 × (12 + 7) = 2 × 19 = 38 mArea: fill the inside.
12 × 7 = 84 m²Notice the units. Perimeter is in m, area in m². Getting the unit right is half the mark, and it is also a check: if you wrote 84 m for the area, the unit itself tells you something is wrong.
A compound shape
Question. An L-shaped room. The whole space would be 10 m × 8 m, but a 4 m × 3 m corner is missing.
Method: subtract the missing piece.
whole: 10 × 8 = 80 m²corner: 4 × 3 = 12 m²
area = 80 − 12 = 68 m²
Alternative method: split into two rectangles and add them. Both give 68 m².
The warning: the perimeter does not shrink the same way. Cutting a corner out of a shape can leave the perimeter unchanged or even longer, because you have removed two edges but added two new ones. Always walk the outline for perimeter, never assume.
Where you need each one
Perimeter: fencing a field, skirting board around a room, a frame for a picture, edging for a flower bed. Anything sold by the metre.
Area: tiles for a floor, paint for a wall, carpet, turf, fabric. Anything sold by the square metre.
A builder who confuses the two will order wildly the wrong amount, and the mistake is expensive in exactly one direction.
A useful check: ask what the material is sold in. Metres → perimeter. Square metres → area.
Where it usually goes wrong
- Mixing up the two. Fence = perimeter, grass = area.
- Wrong units. Area is always squared.
- "Same perimeter means same area." A 1 × 9 rectangle and a 3 × 3 square both have perimeter 20, but areas of 9 and 25.
- Using a slanted side as the height of a triangle.
- Forgetting to convert when the sides are given in different units.
What to remember
- Perimeter = the fence (cm, m) · area = the grass (cm², m²).
- Rectangle: P = 2(l + w), A = l × w.
- Triangle: A = ½ × base × perpendicular height.
- Compound shapes: split or subtract, but walk the outline for perimeter.
Watch one worked through
Here is one done slowly, with the reason for every move. Then one for you.
A room is L-shaped. The full rectangle would be 10 m by 8 m, but a corner of 4 m by 3 m is missing. Find the area of the floor.
- 10 × 8 = 80 m²Find the area of the whole rectangle as if nothing were missing: 10 × 8 = 80 m².A compound shape is easier as one simple shape with a correction than as several pieces counted separately. Fewer steps means fewer places to slip.
- 4 × 3 = 12 m²Find the area of the missing corner: 4 × 3 = 12 m².The cut-out is a rectangle too, so the same length times width rule works on it.
- 80 − 12 = 68 m²Subtract: 80 - 12 = 68 m².You counted the corner once when you took the whole rectangle, so taking it away once leaves exactly the floor that is really there.
- area → m² perimeter → mWrite the unit as m², not m.Area counts squares, so the unit is squared. Perimeter is a distance, so it is not. Marks are lost here more often than on the arithmetic.
Answer: 68 m²
A full rectangle is 9 m by 6 m with a 2 m by 2 m corner missing. What is the area in m²?
Show the answer
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