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 centimeters or meters, never in square units. Watch for that unit whenever you check an answer: it is the fastest way to catch a perimeter and an area that have been swapped.

When a shape has equal or repeated sides, the sum has 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 centimeter on each side has an area of one square centimeter, written $1 \text{ cm}^2$. The area of any region is how many unit squares’ worth of space it covers. For a rectangle the squares tile it exactly, whole and with no gaps; for a shape with a slanted or curved edge, some of the squares along the boundary get cut into pieces, and the area still counts up to how much of a unit square’s worth of space each piece is.

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 gives its area formula. Take any triangle, make an identical copy, turn the copy upside down, and fit the two together along a matching side. The two triangles form 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, and where the height lands depends on the shape. Most often it falls inside the triangle and is not one of the three sides at all; you must use that straight-across height to the base, never the slanted side leaning up to the top corner. If the base meets the opposite side at a right angle, the triangle is a right triangle, and that side already is the height, since it is already perpendicular to the base; there is nothing left to drop.

A third case is easy to miss. If the corner next to the base you chose is obtuse (wider than a right angle), dropping a straight-down line from the far corner misses the base and lands past it, on the line the base sits on. The height is still that straight-down distance, measured at a right angle to the base’s line; you are just measuring to a point beyond the shape’s own edge instead of inside it.

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.

So for a trapezoid,

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.

The perimeter of a composite figure is a different walk. Trace only the outside edge of the shape, the same boundary you would fence. The dashed line that splits the L-shape into two rectangles is a cut you made to find the area; it sits inside the figure, so it is never part of the perimeter.

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 $100$ centimeters in a meter, but a square meter is not $100$ square centimeters. A square meter is a square one meter 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.

Here is why that order matters. A tile is $250 \text{ cm}$ long and $1.2 \text{ m}$ wide. Multiplying the numbers as given, $250 \times 1.2 = 300$, answers no real question, because the two lengths are not in the same unit. Convert first: $250 \text{ cm} = 2.5 \text{ m}$. Now both lengths are in meters, so

Working a formula backward

Every area formula so far has run forward: you knew the lengths, and you multiplied to find the area. Sometimes it runs the other way. You know the area and one length, and a different length is what you need. Each area formula still works; you just treat it as an equation for the unknown length and undo the operations from the outside in, the same way you already undo a two-step equation: undo whichever operation was done to the unknown last, first.

Common mistakes