Imagine you are fencing a garden and then laying turf inside it. The fence follows the edge — that length is the perimeter. The turf fills the middle — that quantity is the area. Two completely different questions, two different units, one shape. Once you can feel the difference, every formula in this lesson stops being something to memorise and starts being something you can simply see.
Perimeter: walk around the edge
Perimeter is the total distance around the outside of a flat shape. To find it, you do the most honest thing imaginable: you walk the boundary and add up every side.
If a triangle has sides of 5 cm, 6 cm and 7 cm, its perimeter is simply \(5 + 6 + 7 = 18\) cm. There is no special trick — perimeter is a length, so its unit is a plain length unit like centimetres (cm) or metres (m).
In words Perimeter answers "how far is it all the way around?" Because it is a distance, the answer is always a length — never a square unit.
Area: count the squares inside
Area answers a different question: "how much flat space is shut inside the boundary?" We measure that by choosing a tile — a square one unit wide and one unit tall — and asking how many of those tiles fit inside without gaps or overlaps.
That tile is why area is measured in square units. One centimetre square is written \(1\text{ cm}^2\); it is a square that is 1 cm on each side. The little raised 2 is not decoration — it is telling you the unit is two-dimensional, a square, not a line.
Drag the steppers below to change a rectangle's width and height. Watch the unit squares fill the inside (that count is the area) while the coloured edge is the distance around (the perimeter). Try to predict both numbers before you read them off.
The rectangle: multiply instead of count
Counting squares one by one works, but it is slow. Look at a rectangle that is 6 units long and 4 units tall: the squares line up in 4 neat rows, with 6 squares in each row. Instead of counting all 24, you can just multiply the rows by the columns. That is the whole secret of the area formula.
For a rectangle with length \(l\) and width \(w\):
\[ A = l \times w \qquad P = 2(l + w) \]The perimeter formula comes from the same honest walk: a rectangle has two lengths and two widths, so the distance around is \(l + w + l + w\), which we tidy up to \(2(l+w)\).
A square is just the special rectangle where the length and width are equal. If each side is \(s\), then \(A = s \times s = s^2\) and \(P = 4s\). You do not need new rules — the rectangle rules already contain it.
- Area is length times width: \(A = 7 \times 3 = 21\). Because we multiplied two lengths in cm, the unit is cm².
- Perimeter is the distance around: \(P = 2(7 + 3) = 2 \times 10 = 20\). This is a single length, so the unit is cm.
The rectangle has an area of 21 cm² and a perimeter of 20 cm. Notice the two answers are different numbers and different units — that is the whole point.
Why a triangle is half a rectangle
Here is the idea that makes triangle area feel obvious instead of arbitrary. Take any triangle, and draw the smallest rectangle that wraps snugly around it — same base along the bottom, same height to the top. The triangle always covers exactly half of that rectangle.
You can see why: the two leftover orange corners can be slid together to make a copy of the very same triangle. So the rectangle is two triangles, and one triangle is half of it. Since the rectangle's area is \(\text{base} \times \text{height}\), the triangle's area is half of that:
\[ A = \tfrac{1}{2} \times \text{base} \times \text{height} \]One warning that trips people up: the height is the straight-up distance from the base to the top point, measured at a right angle to the base. It is not the length of a slanted side. Always use the perpendicular height.
- Start with the surrounding rectangle: \(8 \times 5 = 40\) cm².
- The triangle is half of it: \(A = \tfrac{1}{2} \times 8 \times 5 = \tfrac{1}{2} \times 40 = 20\).
The triangle's area is 20 cm². A good habit: do the \(\text{base} \times \text{height}\) part first, then halve — the arithmetic stays simple.
The parallelogram: lean a rectangle over
A parallelogram looks like a rectangle that has been pushed sideways. Imagine slicing a right-angled triangle off one slanted end and sliding it across to the other end — the leftover pieces snap into a perfect rectangle with the same base and the same height. Nothing was added or removed, so the area is unchanged:
\[ A = \text{base} \times \text{height} \]Just like the triangle, the height here is the perpendicular distance between the two parallel bases — the straight-up gap, not the slanted side. A parallelogram with a base of 6 cm and a height of 4 cm has an area of \(6 \times 4 = 24\) cm², exactly like the rectangle it can be reshaped into.
Composite shapes: cut, solve, add
Real shapes are rarely a single tidy rectangle. An L-shaped room, a cross, a house outline — these are composite shapes, made of simpler pieces stuck together. The strategy never changes: chop the shape into rectangles (and triangles), find each piece's area, then add the pieces up.
Tip — draw the cut line. Before any arithmetic, sketch a single line that splits the shape into two rectangles. Then label each rectangle's own length and width. Most composite-area mistakes come from grabbing the wrong side length, and a clear cut line stops that.
Mind your units
The single most common slip is mixing up length and area. Keep these straight and you will rarely go wrong:
- Perimeter is a length: add the sides, answer in cm (or m, km…).
- Area is squares: it always carries a squared unit, cm² (or m², km²…).
- Every length in a single calculation must be in the same unit before you multiply.
Practice
Try each one yourself, then reveal the full solution.
1. A rectangle is 7 cm long and 3 cm wide. Find its area and its perimeter.
Area is length times width: \(A = 7 \times 3 = 21\) cm².
Perimeter is the distance around: \(P = 2(7 + 3) = 2 \times 10 = 20\) cm.
Area = 21 cm², perimeter = 20 cm.
2. A triangle has a base of 8 cm and a perpendicular height of 5 cm. Find its area.
Use \(A = \tfrac{1}{2} \times \text{base} \times \text{height}\).
First do base times height: \(8 \times 5 = 40\). Then halve it: \(\tfrac{1}{2} \times 40 = 20\).
The area is 20 cm².
3. An L-shape is made from two rectangles: a tall piece 3 cm wide and 8 cm high, with a foot 5 cm wide and 2 cm high attached at the bottom. Find its total area.
Split the L into its two rectangles and find each area separately.
Tall piece: \(3 \times 8 = 24\) cm². Foot: \(5 \times 2 = 10\) cm².
Add the pieces: \(24 + 10 = 34\).
The total area is 34 cm².