Area and Perimeter Formulas: A Worked Breakdown

A practical example: you need to work out how much material goes into a garden bench, and the dimensions off the drawing are in millimetres, 3500 by 3400. The first reaction is to multiply and get an answer. That is exactly what happens, and almost every time something is forgotten: the units of the result, and the extra material needed for cutting. Here is what to compute, in what order, so that both the number and the shopping list come out right.

The second most common kind of request looks like this: diameter and thickness of a part. That is no longer a flat shape but a solid of revolution, so it needs volume. Both cases are below, with numbers you can verify on a calculator.

Perimeter and area: different questions

Mixing these two up happens to almost everyone who computes something with a formula for the first time. The difference is in what is being asked.

• the perimeter is the length of the boundary, in other words how many metres of edging, fence, trim or binding the shape needs;
• the area is the number of square units inside, in other words how much paint, wallpaper, tile or sheet of plywood is needed;
• a rectangle of 3 by 4 metres has a perimeter of 14 metres and an area of 12 square metres, and the two cannot be confused once you keep the units in mind;
• perimeter is measured in metres, centimetres or millimetres, area in square metres, square centimetres or square millimetres;
• an answer of 14 after multiplying 3 by 4 means you computed an area, not a perimeter, and the other way round.

That gives a simple rule worth repeating before any calculation. If the text later mentions length, trim or edging, you need the perimeter. If it mentions material, coverage or square metres, you need the area. Sometimes both are needed at once, for instance when ordering fabric for a cover: the area is the consumption, the perimeter is the length of elastic around the edge.

Formulas for the main shapes

Below are the formulas that cover the overwhelming majority of everyday and construction tasks. The notation is standard: a and b are side lengths, h is the height, r is the radius, d is the diameter, and π is the ratio of circumference to diameter.

• rectangle: area S = a · b, perimeter P = 2 · (a + b);
• square: area S = a², perimeter P = 4 · a;
• circle: area S = π · r², perimeter (circumference) C = 2 · π · r = π · d;
• triangle: area S = ½ · a · h, perimeter P = a + b + c;
• trapezoid: area S = ((a + b) / 2) · h, perimeter P = a + b + c + d, the sum of all four sides;
• parallelogram: area S = a · h, perimeter P = 2 · (a + b);
• rhombus: area S = ½ · d₁ · d₂, perimeter P = 4 · a, where d₁ and d₂ are the diagonals.

Notice that the height appears in almost every area formula, rather than a second side. That is not a typo: a triangle, a trapezoid and a parallelogram can all have sides of different lengths, and the area is determined by the base and the perpendicular height dropped onto it.

Area of a rectangle: where a × b comes from

The formula S = a · b is usually handed over without a proof, and it looks obvious only because it is applied to problems about squared paper. Let us take it apart step by step so it stops looking like magic and becomes a count of squares.

Suppose a rectangle with sides a and b is divided into squares of side 1. There are a columns across and b rows down, so there are a · b small squares. That is the area: how many unit squares fit inside the shape. Changing the size of the small square only changes the unit of measurement, not the substance. The same 3 by 4 metre rectangle holds twelve squares with a side of one metre, twelve hundred squares with a side of ten centimetres and twelve thousand squares with a side of one centimetre, and the area is always 12 m².

Dimensions: 3500 mm = 3.5 m,  3400 mm = 3.4 m

Area:
  S = a · b = 3.5 · 3.4 = 11.9 m²
  in square millimetres: 3500 · 3400 = 11 900 000 mm² = 11.9 m²
  (1 m² = 10 000 cm² = 1 000 000 mm²)

Perimeter:
  P = 2 · (3.5 + 3.4) = 2 · 6.9 = 13.8 m

With a 5 % allowance for cutting:
  11.9 · 1.05 = 12.495 m²   → take 12.5 m²
  13.8 · 1.05 = 14.49 m     → take 14.5 m

Look at how easily the units can go wrong. Substituting 3500 and 3400 as metres into the formula gives 11 900 000 m², and a mistake of that kind is expensive. The reverse happens too: dimensions in centimetres while the material is sold in square metres, and then the area has to be divided by 10 000. The area calculator runs the same computation without that risk: it prints the formula and the steps, and for a circle it accepts a radius, a diameter or a circumference as input.

The circle: πr² and what π is

The area of a circle equals π times the square of the radius. That is short enough, but two things hide in this formula and are worth understanding: what π is, and why it is the square of the radius rather than the radius itself.

π is the ratio of the circumference to the diameter, and it is the same for every circle regardless of size. The ratio of circumference to diameter sits between 3.14159 and 3.14160, and calculations use 3.14159... The point of the formula is that the area of a circle scales with the square of the radius: double the radius and the area grows fourfold, not twofold. The square law here is the same as in a rectangle, only there the two lengths are different, while here the same length appears twice.

Circle with radius 2 m:
  S = π · r² = 3.14159 · 4 ≈ 12.566 m²
  C = 2 · π · r = 2 · 3.14159 · 2 ≈ 12.566 m

Circle with diameter 2 m (radius 1 m):
  S = π · 1² ≈ 3.142 m²
  C = π · d = 3.14159 · 2 ≈ 6.283 m

Check: the area of a circle of diameter d equals π · d² / 4
  3.14159 · 4 / 4 = 3.14159 m²   matches

Forgetting to halve the diameter inflates the area fourfold.

The area of a circle is needed rarely, while the circumference is needed constantly: kerbing, hoops, cutting a cake, sizing a reel. And there the radius versus diameter mistake costs half the material.

Triangles: height and the formula of Heron

A triangle has two working area formulas, and they answer different questions. The first, S = ½ · a · h, needs the height. The second, the formula of Heron, works from the three sides and avoids the height altogether, which is handy when there is no way to measure it.

Heron's formula reads S = √(p · (p − a) · (p − b) · (p − c)), where p is the semiperimeter, half the sum of the three sides. It is exactly what the area calculator uses under the hood, and it is worth checking it on the classic right triangle with sides 3, 4 and 5.

Semiperimeter:
  p = (3 + 4 + 5) / 2 = 6

Heron's formula:
  S = √(6 · (6−3) · (6−4) · (6−5))
    = √(6 · 3 · 2 · 1) = √36 = 6

Via the height (leg 3 as the base, height 4):
  S = ½ · 3 · 4 = 6   same answer

Another example: sides 13, 14 and 15
  p = 21
  S = √(21 · 8 · 7 · 6) = √7056 = 84

Before substituting, check that the triangle exists at all: the sum of any two sides must exceed the third. For 3, 4 and 5 that holds, while 3, 4 and 8 do not form a triangle and Heron's formula would produce a square root of a negative number. The area calculator performs exactly this check and reports an error instead of a result.

Area from a diagonal and reverse problems

Sometimes the sides are unknown and the diagonals are known. For a quadrilateral with perpendicular diagonals the formula S = ½ · d₁ · d₂ works, and it is especially convenient for a rhombus, whose diagonals are always perpendicular. When the diagonals are not perpendicular the formula underestimates the area, and the shape is then split into two triangles instead.

A reverse problem comes up even more often: the diagonal and one side are known, and the second side and the area are wanted. That is where the Pythagorean theorem enters, and after that everything is routine.

Second side:
  b = √(d² − a²) = √(4.88² − 3.4²)
    = √(23.8144 − 11.56) = √12.2544 ≈ 3.5006 m

Area:
  S = a · b = 3.4 · 3.5006 ≈ 11.90 m²

Check: the diagonal of the original bench
  d = √(3.5² + 3.4²) = √(12.25 + 11.56) = √23.81 ≈ 4.880 m

Area of a rhombus with diagonals 6 m and 4 m:
  S = ½ · 6 · 4 = 12 m²

Units: millimetres, centimetres, metres

Half of the mistakes in area and perimeter calculations have nothing to do with formulas and everything to do with units. A few relationships are worth knowing by heart, and converting stops being a separate problem.

• 1 cm = 10 mm, 1 m = 100 cm = 1000 mm, 1 inch = 25.4 mm;
• 1 m² = 10 000 cm² = 1 000 000 mm², so converting an area from square millimetres to square metres means dividing by a million;
• 1 hectare = 10 000 m², the unit you see in property listings;
• to convert a length from centimetres to metres divide by 100, from millimetres divide by 1000;
• area and perimeter convert in different powers: length is a metre, area is a square metre, and dividing by 100 applies to neither of them in the same way.

The request about the diameter and the thickness of a part works the same way, with volume added. The most common blank in engineering is a disc, that is a cylinder with a known diameter and thickness.

Radius: r = 40 / 2 = 20 mm

Volume of the cylinder:
  V = π · r² · h = 3.14159 · 400 · 10 ≈ 12 566 mm³
  = 12.566 cm³   (1 cm³ = 1000 mm³)

Mass in steel (density 7.85 g/cm³):
  m = 12.566 · 7.85 ≈ 98.6 g

If the part is steel and the mass is needed in kilograms: 0.0986 kg.

Working in centimetres is convenient whenever the density is given in grams per cubic centimetre: a diameter of 40 mm is 4 cm and a thickness of 10 mm is 1 cm, so the result comes out as π · 2² · 1 = 12.566 cm³ with no intermediate conversions. The formulas for a sphere, a cylinder, a cone, a prism, a pyramid and a rectangular box are collected in the volume calculator, and the unit converter handles lengths and weights between the different systems — the coefficients are covered in more detail in the piece about converting units.

Typical mistakes

• perimeter and area got swapped, and the edging was ordered instead of the material, or the other way round;
• lengths in millimetres went into the formula while the answer was needed in square metres, producing an error of a million;
• the diameter was used where the radius belonged in πr², and the area came out four times too large;
• the allowance for cutting and layout was forgotten, and the material covered the exact number of parts and nothing more;
• for a trapezoid the sloping side was used instead of the height, because the height is perpendicular to the bases while the leg is usually longer;
• π was rounded to 3.14 too early and precision was lost, which is not acceptable in school calculations;
• the dimensions were not converted to a single unit before the calculation started, only afterwards;
• the possibility that the triangle does not exist was ignored, producing an imaginary area.

The last one deserves special attention, because nearly all of the above are caught by a simple check. Take a couple of round numbers and recompute the result by a different route. If the area of a triangle was found with Heron's formula and then again from the height, and the two agree, the work was done right.

Common questions

Area or perimeter: which one do I need for material?

For material bought by area, whether tile, wallpaper, paint or plywood, you need the area plus an allowance for cutting, usually 5 to 10 per cent. You need the perimeter when buying by length: skirting board, kerbing, trim, or fabric with extra for hems.

How do I find the area if I only know two sides and the angle between them?

For a parallelogram the formula is S = a · b · sin α, where α is the angle between the sides. For a triangle it is S = ½ · a · b · sin α. When the angle is 90 degrees the sine is 1 and the formulas turn into the familiar ones. Practical advice: if the angle is only known roughly, measuring the height is cheaper than working with a sine.

How do I find the area of an irregular plot of land?

Split the plot into rectangles, triangles and trapezoids, compute the area of each and add them up. Curved sections are approximated with rectangles or with the formula of a circle for arcs. The usual accuracy of this method is around 1 per cent, which is enough for most practical work.

Why is the area of a 5 by 5 square not 25 centimetres?

It is, but the unit has to be written down. Five centimetres squared is 25 square centimetres, not 25 centimetres. Area is measured in square units, so the word square always appears in the notation. If a result reads simply 25, the unit was lost somewhere on the way, which is a reliable sign of a dimensional mistake.

How do I find the area of a circle when I only know the circumference?

Rearrange C = 2πr to get r = C / (2π). For a circumference of 12.566 m the radius comes out as 2 m and the area as π · 4 ≈ 12.566 m². The area calculator has a dedicated mode that takes the circumference as input and prints the area together with the perimeter.