# Aire et périmètre des figures planes

> Calculez l'aire et le périmètre de 15 figures : rectangles, triangles, trapèzes, cercles, ellipses, polygones réguliers et anneaux, avec dessins à l'échelle.

Version interactive : https://www.calcopenly.com/fr/geometry/shape-area-perimeter-calculator
Sujet : Calculatrices de géométrie

Choose one of 15 plane shapes and enter the dimensions that define it. The area and perimeter come back with the other lengths that shape has: diagonals, heights, arc and chord lengths, a regular polygon's apothem and circumradius, or a ring's width. A triangle from three sides uses Heron's formula, A = √(s(s − a)(s − b)(s − c)) with s half the perimeter, and a regular polygon uses A = ns²/(4 tan(180°/n)).

The default rectangle, 8 cm by 5 cm, has an area of 40 cm², a perimeter of 26 cm and a diagonal of 9.434 cm. Area sizes flooring, turf and paint; perimeter sizes fencing, edging and skirting board. The drawing is to scale, so a mistyped dimension shows up as the wrong shape.

Base and height do not fix a triangle's slanted sides, and a trapezoid needs its legs, so those two inputs give area only. An ellipse's perimeter has no closed form; Ramanujan's second approximation is used, accurate to 0.04% or better.

## Données

- **Shape** (options : Square, Rectangle, Triangle (base and height), Triangle (three sides), Parallelogram, Trapezoid (trapezium), Rhombus, Kite, Cercle, Semicircle, Circular sector, Circular segment, Ellipse, Regular polygon, Annulus (ring))
- **Side length**
- **Number of sides**
- **Longueur**
- **Largeur**
- **Base**
- **Slanted side**
- **Trapezoid defined by** (options : Bases and height, Bases and legs)
- **Bottom base (a)**
- **Top base (b)**
- **Hauteur**
- **Left leg (c)**
- **Right leg (d)**
- **Side a**
- **Side b**
- **Side c**
- **Diagonal p**
- **Diagonal q**
- **Shorter pair of sides (a)**
- **Longer pair of sides (b)**
- **Angle between a and b**
- **Rayon**
- **Central angle**: Entre 0° et 360°
- **Semi-axis a**
- **Semi-axis b**
- **Outer radius (R)**
- **Inner radius (r)**
- **Unité des résultats** (options : Millimètres (mm), Centimètres (cm), Mètres (m), Kilomètres (km), Pouces (in), Pieds (ft), Yards (yd), Miles (mi))

## Résultats

- Aire — résultat principal
- Périmètre
- Diagonal
- Second diagonal
- Side length
- Hauteur
- Longueur de l’arc
- Longueur de la corde
- Apothem (inradius)
- Circumradius
- Ring width

## Formule

$$
\begin{gathered} A_\triangle = \sqrt{s(s-a)(s-b)(s-c)},\quad A_{n\text{-gon}} = \frac{n s^2}{4\tan(\pi/n)} \\ P_\text{ellipse} \approx \pi(a+b)\left(1 + \frac{3h}{10 + \sqrt{4-3h}}\right) \end{gathered}
$$

## Exemples détaillés

### Rectangle 8 × 5 cm

- Shape: Rectangle
- Longueur: 8 cm
- Largeur: 5 cm
- Unité des résultats: Centimètres (cm)
- **Aire: 40 cm²**
- **Périmètre: 26 cm**
- **Diagonal: 9.433981 cm**
- Source de vérification : 8 × 5, 2(8 + 5); Python 3.8 math.sqrt(89)

### 3-4-5 triangle by Heron's formula

- Shape: Triangle (three sides)
- Side a: 3 cm
- Side b: 4 cm
- Side c: 5 cm
- Unité des résultats: Centimètres (cm)
- **Aire: 6 cm²**
- **Périmètre: 12 cm**
- **Hauteur: 4 cm**
- Source de vérification : Right triangle with legs 3 and 4: ½·3·4 = 6; height onto side 3 is the other leg, 4

### Regular hexagon, side 2 m

- Shape: Regular polygon
- Side length: 2 m
- Number of sides: 6
- Unité des résultats: Mètres (m)
- **Aire: 10.392305 m²**
- **Périmètre: 12 m**
- **Apothem (inradius): 1.732051 m**
- **Circumradius: 2 m**
- Source de vérification : Python 3.8 math: 6*4/(4*tan(pi/6)) = 6√3, 2/(2*tan(pi/6)) = √3; a hexagon's circumradius equals its side

### Square as a regular 4-gon (exact tan 45°)

- Shape: Regular polygon
- Side length: 3 cm
- Number of sides: 4
- Unité des résultats: Centimètres (cm)
- **Aire: 9 cm²**
- **Apothem (inradius): 1.5 cm**
- **Circumradius: 2.12132 cm**
- Source de vérification : Square of side 3: area 9, apothem 3/2; Python 3.8 math: 3/sqrt(2)

### Ellipse with a = b is a circle (edge case)

- Shape: Ellipse
- Semi-axis a: 5 cm
- Semi-axis b: 5 cm
- Unité des résultats: Centimètres (cm)
- **Aire: 78.539816 cm²**
- **Périmètre: 31.415927 cm**
- Source de vérification : h = 0 so Ramanujan II reduces to 2πr; Python 3.8 math: 25*pi, 10*pi

### Ellipse 10 × 6 (semi-axes)

- Shape: Ellipse
- Semi-axis a: 10 cm
- Semi-axis b: 6 cm
- Unité des résultats: Centimètres (cm)
- **Aire: 188.495559 cm²**
- **Périmètre: 51.053998 cm**
- Source de vérification : Python 3.8 math: 60*pi and Ramanujan II with h = 1/16; the exact perimeter by the AGM series (Python decimal, 40 digits) is 51.0539977268, 1.2e-9 higher

## Questions

### How do you find the area of a trapezoid?

Average the two parallel sides and multiply by the perpendicular height: A = (a + b)/2 × h. Bases of 10 cm and 4 cm with a height of 4 cm give 7 × 4 = 28 cm². British English calls this shape a trapezium; the formula is the same. The perimeter also needs the two slanted legs: legs of 5 cm each make it 24 cm.

### How do you find the area of a triangle from three sides?

Use Heron's formula. Take the semi-perimeter s = (a + b + c)/2, then A = √(s(s − a)(s − b)(s − c)). Sides of 5, 6 and 7 cm give s = 9 and A = √(9 × 4 × 3 × 2) = √216 ≈ 14.697 cm². Three lengths only make a triangle if each is shorter than the other two combined, so 3, 4 and 8 have no area.

### What is the area of a regular hexagon?

A = (3√3/2)s², about 2.598 times the side squared. A hexagon with 5 cm sides covers 64.952 cm², and one with 2 m sides covers 10.392 m². Any regular polygon with n sides of length s has A = ns²/(4 tan(180°/n)). A hexagon's circumradius equals its side, and its apothem is s√3/2 ≈ 0.866s.

### Is there a formula for the perimeter of an ellipse?

Not an exact one in elementary functions: the exact perimeter is a complete elliptic integral. Ramanujan's 1914 approximation, P ≈ π(a + b)(1 + 3h/(10 + √(4 − 3h))) with h = (a − b)²/(a + b)², is off by at most 0.04%, reached only as the ellipse flattens into a line. For semi-axes of 10 and 6 it gives 51.0539977, within 1.2 × 10⁻⁹ of the exact value.

### How do you convert square feet to square metres?

Multiply by 0.09290304, the square of the international foot (0.3048 m exactly, fixed in 1959). One square metre is about 10.764 ft². A 12 ft × 10 ft room is 120 ft², or 11.148 m². Area scales with the square of the length unit, so converting each side first and then multiplying gives the same result.

### Quelle est la précision de « Aire et périmètre des figures planes » ?

La précision dépend de vos données et des hypothèses de la méthode. Le calcul décimal utilise 50 chiffres significatifs, mais les estimations, méthodes numériques et données sources peuvent être moins précises ; l’arrondi affiché ne supprime pas ces limites. Exemples résolus vérifiés à partir de sources indépendantes : 16. Par exemple, « Rectangle 8 × 5 cm » est vérifié à l’aide de 8 × 5, 2(8 + 5); Python 3.8 math.sqrt(89).

### D’où vient cette méthode ?

Weisstein, E. W. “Heron's Formula”, “Regular Polygon”, “Ellipse” — MathWorld; Ramanujan, S. (1914) “Modular equations and approximations to π”, Quarterly Journal of Mathematics 45, 350–372 (perimeter approximation II); NIST Handbook 44, Appendix C — exact inch–centimetre relation (1 in = 2.54 cm).

## Sources

- [Weisstein, E. W. “Heron's Formula”, “Regular Polygon”, “Ellipse” — MathWorld](https://mathworld.wolfram.com/Ellipse.html)
- Ramanujan, S. (1914) “Modular equations and approximations to π”, Quarterly Journal of Mathematics 45, 350–372 (perimeter approximation II)
- [NIST Handbook 44, Appendix C — exact inch–centimetre relation (1 in = 2.54 cm)](https://www.nist.gov/pml/owm/nist-handbook-44-current-edition)