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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.

Mis à jour Exemples vérifiés : 16

Essayer
Aire
cm²
Aire: 40 cm²
Décimales maximales : 4 ; Au plus proche, égalités en s’éloignant de zéro
Périmètre
26cm
Diagonal
9.434cm

The rectangle covers 40 cm² and its boundary is 26 cm long.

The rectangle, to scale

l = 8 cmw = 5 cm
Comment le calcul est effectué S

  1. Aire

    A=l×w=8×5=40 cm2A = l \times w = 8 \times 5 = 40\,\text{cm}^{2}

  2. Périmètre

    P=2(l+w)=2(8+5)=26 cmP = 2(l + w) = 2(8 + 5) = 26\,\text{cm}

  3. Diagonal

    d=l2+w2=82+52=9.433981 cmd = \sqrt{l^2 + w^2} = \sqrt{8^2 + 5^2} = 9.433981\,\text{cm}

À propos de Aire et périmètre des figures planes

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.

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)

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).

À propos de ce calculateur

A△=s(s−a)(s−b)(s−c),An-gon=ns24tan⁡(π/n)Pellipse≈π(a+b)(1+3h10+4−3h)\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}

Sources

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

Vérifié avec les références

Ce calculateur comprend 16 exemples résolus dont les réponses proviennent de sources indépendantes. Ils font partie de la suite de tests et peuvent aussi être exécutés ici.

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