Perimeter and Area
Wrapping a ribbon all the way around a present and then covering the whole box in paper are
two very different jobs: the ribbon follows the edge right around, while the paper
has to fill the space inside.
Every flat shape has two important measurements, and they are easy to muddle up. One is how
far it is around the edge. The other is how much space it covers
inside. The first is called the perimeter; the second is the
area. This page is all about telling them apart.
Perimeter: the distance around
The perimeter of a shape is the total distance all the way around its edge.
Imagine a tiny ant walking around the outside of the shape until it gets back to where it
started — the length of that whole journey is the perimeter. To find it, just walk around
and add up every side length as you pass it.
A rectangle has two long sides and two short sides. If the length is
l and the width is w, there are two of
each, so a quick way to add them all is:
P = l + w + l + w = 2(l + w)
For any other shape there is no shortcut — you simply add every side together. A
triangle with sides 3, 4 and
5 has perimeter
3 + 4 + 5 = 12. Because perimeter is a length, we measure it in
length units like centimetres (cm) or metres (m).
A farmer wants to put a fence right around a field to keep the cow safe. How much fence
should she buy? That is a perimeter question — the fence only goes
around the edge, so she adds up the four sides. If the field is
30 m long and 20 m wide, the fence
must be 2(30 + 20) = 100 m long. The grass inside does not
matter one bit for the fence; only the journey around the edge counts.
Area: the space inside
The area of a shape is how much flat space it covers. We measure it by
covering the inside with little squares that are 1
cm wide and 1 cm tall — a square centimetre,
written 1\ \text{cm}^2 — and counting how many fit.
A rectangle makes this lovely and easy. The squares line up in a neat grid:
l squares along each row, and w rows
stacked up. Instead of counting them one by one, we just multiply:
A = l \times w
So a rectangle 5 cm by 3 cm holds
5 \times 3 = 15 little squares — its area is
15\ \text{cm}^2. Notice the small 2 in
\text{cm}^2: it reminds us we are counting squares, not a single
straight distance.
A builder is tiling a kitchen floor with square tiles. How many tiles will he need? That is
an area question — the tiles cover the whole space inside the
room, not just the edge. If the floor is 4 tiles long and
3 tiles deep, he needs
4 \times 3 = 12 tiles to fill it. Count the rows, count along
each row, and multiply — exactly like the cat's floor below your feet.
Three quick worked examples
Watch how the two measurements come from the same picture but answer different questions:
-
A rectangle 6 cm by 4 cm:
perimeter 2(6 + 4) = 20 cm, area
6 \times 4 = 24\ \text{cm}^2.
-
A square with side 5 cm (all four sides equal):
perimeter 4 \times 5 = 20 cm, area
5 \times 5 = 25\ \text{cm}^2.
-
A long thin rectangle 5 cm by 1 cm:
perimeter 2(5 + 1) = 12 cm, area
5 \times 1 = 5\ \text{cm}^2 — same perimeter as a
3 \times 3 square, but much less space inside!
The two classic perimeter-and-area traps:
-
Different units. Perimeter is a length, so it is measured in
cm (centimetres). Area is squares, so it is measured in
cm² (square centimetres). An answer like "the area is 15 cm" is wrong —
area always wears the little {}^2.
-
Same perimeter does not mean same area. A
5 \times 1 rectangle and a
3 \times 3 square both have perimeter
12 cm, yet their areas are
5\ \text{cm}^2 and 9\ \text{cm}^2.
The edge can be the same length while the space inside is very different.
See it: count the squares, walk the edge
Here is a rectangle drawn on a grid of 1 cm squares. Count along
the bottom to read the length, count up the side to read the
width, then check both answers below: the perimeter walks
the edge, and the area counts every square inside. Press
Refresh for a brand-new rectangle.
A triangle is half a rectangle
Triangles follow the same idea with one twist. Box a triangle inside a rectangle and you will
see it fills exactly half of it — so its area is half of base times height.
Step through the figure to watch the rectangle, then the triangle hiding inside it.
A_{\text{tri}} = \tfrac{1}{2} \times b \times h