Area and perimeter

The difference in one picture#

Picture a plot of land. A fence goes along the boundary — its length is the perimeter. Grass is sown inside — the size of the lawn is the area. You measure a fence in metres and buy turf in square metres. Both quantities describe the same shape but answer different questions: "how far is it all the way round" and "how much fits inside".

The units check always works. If your answer to a question about area is in plain centimetres, you calculated a perimeter. If the answer about the length of a fence is in square metres, you used the wrong formula.

The formulas you need at school#

A rectangle with sides $a$ and $b$: perimeter $P = 2(a+b)$, area $A = ab$.

A square with side $a$: perimeter $P = 4a$, area $A = a^2$.

A triangle: the perimeter is the sum of the three sides, the area is $A = \frac{1}{2}bh$, where $h$ is the height drawn to the base $b$. The height has to be perpendicular to the very side you took as the base.

A circle of radius $r$: circumference $C = 2\pi r$, area $A = \pi r^2$. For a circle the boundary is called the circumference rather than the perimeter, but the idea is the same. For estimates use $\pi \approx 3.14$.

Problem 1: finding both#

A rectangle has sides of 6 cm and 4 cm.

Step 1. The perimeter is the sum of all four sides. Opposite sides are equal, so add the length and the width and double it: $2 \cdot (6 + 4) = 20$ cm.

Step 2. The area is how many 1 cm squares fit inside. There are 6 in each row and 4 rows: $6 \cdot 4 = 24$ cm².

Step 3. Write the units. Perimeter 20 cm, area 24 cm² — different numbers and different units for the same shape.

Problem 2: from perimeter back to area#

A square has a perimeter of 36 cm. Find its area.

Step 1. A square has four equal sides, so $4a = 36$.

Step 2. Divide both sides by 4: $a = 9$ cm. This is an ordinary equation with one unknown, and the moves from how to solve linear equations apply literally.

Step 3. The area of a square is the side squared: $9^2 = 81$ cm².

A common mistake in this problem is to square the perimeter itself and get 1296. You square the side, and you have to find the side first.

Problem 3: a triangle#

The base is 10 cm and the height to it is 6 cm.

Step 1. The area of a triangle is half the base times the height.

Step 2. Work out the product: $10 \cdot 6 = 60$.

Step 3. Halve it: $A = 30$ cm². You take half because the triangle is exactly half of a rectangle with sides 10 and 6.

You cannot find the perimeter of this triangle from these data: the height is not a side, and the lengths of the other two sides are unknown. "Base and height given" answers only the question about area.

Why equal perimeters give different areas#

Take a 6 × 4 rectangle and an 8 × 2 rectangle. The perimeters are the same: $2 \cdot (6+4) = 20$ and $2 \cdot (8+2) = 20$. The areas differ: 24 cm² and 16 cm². The more stretched the rectangle is for the same perimeter, the less fits inside. The largest area for a given perimeter belongs to the square: with a perimeter of 20 cm that is a 5 × 5 square with an area of 25 cm².

The practical lesson for problems about plots and rooms: "the perimeters are equal" does not mean "the areas are equal", and vice versa. If a problem asks by what percentage one area is larger than another, the usual arithmetic from percentage word problems takes over.

Mistakes and how to catch them#

Adding the length and the width and calling it the perimeter: $6 + 4 = 10$ is half the perimeter, and you still need to double it.

Multiplying all four sides to get the area. The area of a rectangle is the product of two adjacent sides, not four.

Taking the height to one side of a triangle and multiplying it by another side. The base and the height are a pair; they must be perpendicular to each other.

Mixing units: one side in metres, the other in centimetres. Convert everything to the same unit before calculating. And remember that 1 m² is 10,000 cm², not 100: when you convert areas, the conversion factor is squared.

Step-by-step plan

1. Step 1 — separate the ideasOn five shapes, point to the boundary and to the inside, and name the unit for each quantity.
2. Step 2 — rectangle and squareFind P and A for ten pairs of sides, writing units in every answer.
3. Step 3 — reverse problemsFind a side from the perimeter, or the second side from the area, each time through an equation.
4. Step 4 — triangle and circleArea from base and height, circumference and area of a circle with π ≈ 3.14.
5. Step 5 — word problemsFences, tiles, wallpaper, lawns: decide 'boundary or inside' before choosing a formula.

Check yourself

Sources

• Khan Academy — GeometryArea and perimeter lessons with practice problems and explanations
  free
• OpenStax Prealgebra 2eFree textbook; the geometry chapter covers perimeter, area and circles with answers
  free

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