Area & Perimeter Calculator

Background.

The Quanta area and perimeter calculator computes both numbers — the area enclosed inside a shape and the length around its edge — for the six most common two-dimensional figures in geometry: rectangle, square, right triangle, parallelogram, trapezoid, and circle. Pick a shape from the dropdown, type in the dimensions it needs, and read off the two answers instantly.

Although these formulas are taught side by side in every primary-school geometry class, area and perimeter measure fundamentally different things, and confusing them is one of the most common sources of error in construction, gardening, real-estate, and design. Area is a two-dimensional quantity — it tells you how much flat surface a shape covers, in square units (square metres, square feet, square centimetres). Perimeter is a one-dimensional quantity — it tells you how long the boundary line is, measured in plain linear units (metres, feet, centimetres).

A 10 m × 1 m driveway has a perimeter of 22 m and an area of 10 m². A 4 m × 2.5 m driveway also has an area of 10 m², but a perimeter of 13 m — almost half as much edge for the same surface. Two shapes can have the same area and wildly different perimeters, or the same perimeter and wildly different areas.

The classical statement of that fact is the isoperimetric inequality (Steiner, 1838): of all closed plane curves with a given perimeter P, the circle is the one that encloses the largest area, A ≤ P² ÷ (4π). Equivalently, of all shapes with a given area, the circle has the shortest perimeter. Every other shape — square, rectangle, triangle, ellipse, the irregular plot of land at the back of your house — pays a perimeter premium for the same enclosed area. This is why bubbles are round, why animal cells are round, why insulated water tanks are cylindrical, and why a long thin rectangle is always a bad shape for a fenced yard if you want to minimise fencing cost per square metre of grass.

Below the calculator we set out each of the six formulas with its derivation, walk through a worked example with a 3-4-5 right triangle (the smallest Pythagorean triple — A = 6, P = 12), state explicitly the two simplifying assumptions the parallelogram and trapezoid perimeters use (the formula treats the slant sides as if they were vertical, which is exact for rectangles and right trapezoids and an approximation otherwise), answer the questions people search for most around these formulas — including the isoperimetric one — and cite the primary sources: Euclid's Elements Book I, NIST's Digital Library of Mathematical Functions, Steiner's symmetrisation argument, MathWorld, and the CRC Standard Mathematical Tables.

What is area & perimeter calculator?

Area is the measure of a two-dimensional region — the number of unit squares that fit inside the shape. It is always reported in square units: square metres (m²), square feet (ft²), square inches (in²), square centimetres (cm²), hectares, acres, and so on. Perimeter is the measure of a one-dimensional boundary — the total length of the closed curve that bounds the shape. It is reported in plain linear units (m, ft, in, cm). For a circle the perimeter has its own name, circumference, but it is the same idea — once around the edge. The six shapes this calculator handles cover the great majority of everyday geometry problems, and each has a closed-form area and perimeter formula that has been known since antiquity. Rectangle: A = L × W; P = 2(L + W). Square: A = s²; P = 4s. Right triangle: A = ½ × base × height; P = base + height + √(base² + height²), where the square-root term is the hypotenuse via the Pythagorean theorem. Parallelogram: A = base × perpendicular height; P = 2(base + slant side) — and because this calculator only takes base and height as inputs, the perimeter is reported using the rectangle approximation P ≈ 2(base + height), which is exact only when the parallelogram is a rectangle and otherwise a lower bound. Trapezoid: A = ½ × (base₁ + base₂) × height; P = base₁ + base₂ + leg₁ + leg₂ — and again, because only the two bases and the perpendicular height are taken as inputs, the perimeter is reported using the right-trapezoid approximation P ≈ base₁ + base₂ + 2 × height, which is exact only when both legs are perpendicular to the bases. Circle: A = πr²; P (circumference) = 2πr. These six formulas, between them, will solve every plane-area problem a homeowner, builder, designer, or student is likely to encounter.

How to use this calculator.

1. 01Choose the shape from the dropdown. The calculator supports six: Rectangle, Square, Right Triangle, Parallelogram, Trapezoid, and Circle. Each shape uses a different subset of the input fields, and the irrelevant fields are simply ignored.
2. 02For a Rectangle, type Length (the longer side) and Width (the shorter side). For a Square, type the Side length. For a Right Triangle, type the Length (base) and Width (height — the leg perpendicular to the base). The hypotenuse is computed automatically using the Pythagorean theorem, so you do not need to enter it.
3. 03For a Parallelogram, type the Length (base) and Width (perpendicular height, not the slant side). The area is exact; the perimeter uses the rectangle approximation P ≈ 2(base + height) because the slant length is not requested as a separate input — see the formula section below for the assumption.
4. 04For a Trapezoid, type the Length (base₁, the longer parallel side), Base 2 (base₂, the shorter parallel side), and Width (the perpendicular distance between the two parallel sides). The area is exact. The perimeter uses the right-trapezoid approximation P ≈ base₁ + base₂ + 2 × height — exact only when both legs are perpendicular to the bases, otherwise a lower bound.
5. 05For a Circle, type the Radius. The calculator returns the area (πr²) and the circumference (2πr) — the circle's 'perimeter'. If you only know the diameter, divide by 2 first. If you only know the circumference, divide by 2π ≈ 6.2832.
6. 06Read the two outputs. Area is the primary result, reported in 'sq units' — apply your own square label (sq m, sq ft, sq in, etc.). Perimeter is the secondary result, reported in the same linear unit you typed.
7. 07Keep units consistent. The calculator does not convert between units — if you enter the length in metres and the width in centimetres, the answer is meaningless. Pick one unit (everything in m, or everything in cm, or everything in ft) and stick with it for every field.

The formula.

Six shapes, six pairs of formulas. Throughout, units must be consistent — area comes out in the square of whatever linear unit you typed.

RECTANGLE. A = L × W (length times width — the count of unit squares that tile the inside). P = 2(L + W) (two long sides plus two short sides). The formulas are essentially axiomatic in Euclidean geometry — they are the basis on which every other planar area is defined.

SQUARE. A = s² (side squared — a square is a rectangle with L = W = s). P = 4s. A square is the rectangle of given perimeter that encloses the maximum area, and the rectangle of given area that has the minimum perimeter — the rectangular case of the isoperimetric inequality.

RIGHT TRIANGLE. A = ½ × base × height — a right triangle is exactly half of a rectangle of the same base and height, sliced along the diagonal. P = base + height + hypotenuse, where hypotenuse = √(base² + height²) by the Pythagorean theorem (Euclid, Elements, Book I, Proposition 47). The famous 3-4-5 triangle has area ½ × 3 × 4 = 6 and perimeter 3 + 4 + 5 = 12.

PARALLELOGRAM. A = base × perpendicular height. This is exact, and is the same arithmetic as a rectangle — geometrically, you can slice a triangular wedge off one end of a parallelogram and slide it to the other end to turn it into a rectangle of the same base and same perpendicular height. P, however, is not 2(base + height) for a generic parallelogram — the true perimeter is 2(base + slant), where the slant is the length of the non-base side, related to base and height by slant ≥ height (with equality only when the parallelogram is a rectangle). ASSUMPTION USED BY THIS CALCULATOR: because the slant length is not requested as a separate input, the engine reports P = 2(base + height) — the rectangle case, which is exact for rectangles and a lower bound for every other parallelogram. If you need the exact perimeter of a non-rectangular parallelogram, compute the slant separately (slant² = height² + offset², where offset is the horizontal shear of the top side relative to the bottom) and add 2 × slant + 2 × base by hand.

TRAPEZOID. A = ½ × (base₁ + base₂) × perpendicular height. This is exact — geometrically, a trapezoid has 'average base × height' as its area, where the average is (base₁ + base₂) / 2. P, however, equals base₁ + base₂ + leg₁ + leg₂ in general, and the two legs are not equal to the height unless the trapezoid is a right trapezoid (both legs perpendicular to the bases). ASSUMPTION USED BY THIS CALCULATOR: because only the two bases and the perpendicular height are taken as inputs, the engine reports P ≈ base₁ + base₂ + 2 × height — the right-trapezoid case. This is exact when both legs are vertical and a lower bound otherwise. For an isosceles or general trapezoid where you know the leg length, compute the true perimeter as base₁ + base₂ + leg₁ + leg₂ by hand.

CIRCLE. A = πr² (Archimedes, Measurement of a Circle, ~250 BCE). P (circumference) = 2πr — π is defined as the ratio of any circle's circumference to its diameter, so C = π × d = 2πr by direct substitution. The calculator uses JavaScript's Math.PI, which is accurate to roughly 15 significant decimal digits.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Euclid, Elements, Book I, Propositions 35–47 (~300 BCE) — the foundational propositions on plane rectilinear figures, including parallelogram area equivalence (Prop 35), triangle area equivalence (Prop 37), and the Pythagorean theorem (Prop 47), which underpins the right-triangle hypotenuse used in the perimeter formula. David E. Joyce annotated edition, Clark University.
2. [2]Archimedes, Measurement of a Circle (~250 BCE) — the original rigorous proof that the area of a circle is πr² and the circumference is 2πr, plus the first rigorous bounds on π between 223/71 and 22/7. Translation by T. L. Heath, The Works of Archimedes (Cambridge University Press, 1897).
3. [3]Steiner, Jacob. 'Einfache Beweise der isoperimetrischen Hauptsätze' (1838) — Steiner's symmetrisation argument for the isoperimetric inequality, showing that among all plane shapes of a given perimeter, the circle encloses the largest area. Reprinted in Gesammelte Werke, Vol. 2 (Reimer, 1882).
4. [4]NIST Digital Library of Mathematical Functions, §4.14 (Trigonometric Functions) and §1.6 (Vectors and Vector-Valued Functions) — reference definitions of π and the circle/disc area integrals used in the formula derivations.
5. [5]Weisstein, Eric W. 'Isoperimetric Inequality.' From MathWorld — A Wolfram Web Resource. Comprehensive modern reference for the inequality P² ≥ 4πA, including history (Dido's problem, Steiner symmetrisation) and proofs.
6. [6]Weisstein, Eric W. 'Trapezoid.' From MathWorld — A Wolfram Web Resource. Standard reference for the trapezoid area formula A = ½(b₁ + b₂)h and the distinction between general, isosceles, and right trapezoids relevant to the perimeter approximation used here.
7. [7]Weisstein, Eric W. 'Parallelogram.' From MathWorld — A Wolfram Web Resource. Reference for A = bh exactness and the slant-side dependence of the perimeter formula, which motivates the rectangle approximation used in this calculator.
8. [8]Zwillinger, Daniel (ed.). CRC Standard Mathematical Tables and Formulae, 33rd edition. CRC Press, 2018 — Chapter 4 (Geometry) contains the canonical area and perimeter formulas for all six shapes covered by this calculator, with cross-references to derivations.