Perimeter and Area

Perimeter: the distance around

The perimeter of a shape is the total distance around its boundary. For any polygon (a shape with straight sides) there is nothing to memorize: walk around the edge and add up the side lengths. If a triangle has sides $5$ cm, $6$ cm, and $7$ cm, its perimeter is

Because perimeter is a sum of lengths, it is itself a length. So perimeter is measured in linear units like centimetres or metres, never in square units. That single fact keeps perimeter and area from ever being confused.

When a shape has equal or repeated sides, the sum collapses into a shortcut. A rectangle has two sides of length $l$ and two of width $w$, so

A square is a rectangle whose four sides are all the same length $s$, so the sum is just $s$ four times:

These are not new rules, only the side-adding rule written compactly for a shape whose sides repeat.

Area: counting unit squares

Area measures how much flat space a shape covers. To measure it we need a fixed unit, and the natural choice is a unit square: a square one unit long on every side. A square one centimetre on each side has an area of one square centimetre, written $1 \text{ cm}^2$. The area of any region is simply the number of unit squares it takes to tile the region with no gaps and no overlaps.

This is why area is always measured in square units. A length is measured by laying a ruler along a line; an area is measured by laying unit squares across a surface. A surface needs two directions to cover it, and that is exactly what the little raised $2$ in $\text{cm}^2$ records.

Area of a rectangle

Counting squares one at a time would be slow for a large shape, so we look for a pattern. Tile a rectangle that is $5$ units wide and $3$ units tall with unit squares, as in the figure above. The squares fall into a neat array: $3$ rows, each holding $5$ squares. The total is then a multiplication, not a long count:

That is the whole idea behind the rectangle formula. The width tells you how many squares sit in each row, and the height tells you how many rows there are. So multiplying the two numbers counts every square exactly once.

So for a rectangle,

A square is a rectangle whose length and width are the same value $s$, so its area is

This is exactly why $s^2$ is read ”$s$ squared”. Raising a length to the second power is the area of the square built on that length. The exponent notation you met earlier and the geometry of a square are the same idea seen from two sides.

Perimeter and area are easy to mix up because both are called “how big it is”. The surest way to keep them apart is to change a rectangle and watch what each one does. Set the width and the height below, and both numbers are reported for the shape you built.

Here is the experiment worth doing. Build the $4$ by $4$ square: its perimeter is $16$ and its area is $16$. Now build the rectangle $7$ wide and $1$ tall: its perimeter is still $16$, but its area has fallen to $7$. Same distance around, less than half the surface inside. Perimeter measures the fence and area measures the field, and knowing one tells you very little about the other.

Area of a parallelogram

A parallelogram is a four-sided shape whose opposite sides are parallel. It looks like a rectangle that has been pushed over so it leans. You might guess that its area is the product of two of its sides, but that is the classic trap. The figure shows why: the slanted side is longer than the straight-up distance across the shape, so multiplying by it would overcount.

The quantity that actually matters is the height, meaning the perpendicular distance between the base and the side opposite it. That distance is measured straight across at a right angle, not along the slanted edge. With the base called $b$ and that perpendicular height called $h$, the area turns out to be exactly $b \times h$. We can see why by turning the parallelogram into a rectangle.

So for a parallelogram,

A rectangle is just a parallelogram that happens to stand up straight. In that case the height equals the vertical side, and the formula collapses back to $A = l \times w$.

Area of a triangle

A triangle is half of a parallelogram, and that single observation hands us its area. Take any triangle, make an identical copy, turn the copy upside down, and fit the two together along a matching side. The two triangles snap into a parallelogram with the same base $b$ and the same perpendicular height $h$ as the original triangle.

So for a triangle,

The perpendicular-height warning matters even more for triangles than for parallelograms. The reason is that a triangle’s height often falls inside the shape and is not one of the three sides at all. You must use the straight-across height to the base, never the slanted side leaning up to the top corner.

Area of a trapezoid

A trapezoid is a four-sided shape with at least one pair of parallel sides. Call those two parallel sides $a$ and $b$, and call the perpendicular distance between them the height $h$. When the two parallel sides have different lengths the trapezoid is wider at one than the other, so neither length alone gives the area. What works is the average of the two parallel sides, multiplied by the height. Taking “at least one pair” rather than “exactly one pair” lets a parallelogram count as a special trapezoid whose two parallel sides are equal. You will see in a moment that the trapezoid formula then collapses neatly onto both the parallelogram formula and the triangle formula.

So for a trapezoid,

This one formula quietly contains the others. If the two parallel sides are equal ($a = b$), the trapezoid is a parallelogram and the formula gives $\tfrac{1}{2}(a + a)h = a h$, matching the parallelogram rule. If the top side shrinks to nothing ($a = 0$), the shape becomes a triangle and the formula gives $\tfrac{1}{2}(0 + b)h = \tfrac{1}{2} b h$, matching the triangle rule. Every area rule in this lesson is one idea seen at different settings.

Composite figures: cut into pieces you know

Many real shapes are not a single tidy rectangle or triangle, but you can almost always decompose them. Cut the figure into rectangles and triangles, find each piece’s area with the rules above, and then add the pieces together. Area is additive, so the area of the whole is the sum of the areas of the non-overlapping parts.

The L-shape above is split into a tall rectangle on the left and a short one on the right. The left piece is $6 \text{ m}$ by $9 \text{ m}$, giving $54 \text{ m}^2$; the right piece is $5 \text{ m}$ by $5 \text{ m}$, giving $25 \text{ m}^2$; together the figure covers $54 + 25 = 79 \text{ m}^2$. Splitting a different way, or subtracting the missing corner from a full rectangle, gives the same total, because the actual region covered has not changed.

A caution about units

Perimeter and area use different kinds of units, and mixing them up is the most common error in the whole topic. Perimeter is a length, so it is reported in linear units: cm, m, km. Area is a count of unit squares, so it is reported in square units: $\text{cm}^2$, $\text{m}^2$, $\text{km}^2$. A number with no unit, or with the wrong unit, is not a complete answer.

Unit conversions deserve special care, because squaring a length squares the conversion factor too. There are $100$ centimetres in a metre, but a square metre is not $100$ square centimetres. A square metre is a square one metre on each side, which is $100 \text{ cm}$ by $100 \text{ cm}$, so

Always convert lengths to the same unit before computing an area, and remember that the square unit grows by the conversion factor multiplied by itself.

Worked examples

Common mistakes