# Fläche und Umfang ebener Figuren berechnen

> Fläche und Umfang von 15 Figuren: Rechtecken, Dreiecken, Trapezen, Kreisen, Ellipsen, regelmäßigen Vielecken und Ringen, mit maßstabsgetreuer Zeichnung.

Interaktive Version: https://www.calcopenly.com/de/geometry/shape-area-perimeter-calculator
Thema: Geometrierechner

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.

## Eingaben

- **Shape** (Optionen: Square, Rectangle, Triangle (base and height), Triangle (three sides), Parallelogram, Trapezoid (trapezium), Rhombus, Kite, Kreis, Semicircle, Circular sector, Circular segment, Ellipse, Regular polygon, Annulus (ring))
- **Side length**
- **Number of sides**
- **Länge**
- **Breite**
- **Base**
- **Slanted side**
- **Trapezoid defined by** (Optionen: Bases and height, Bases and legs)
- **Bottom base (a)**
- **Top base (b)**
- **Höhe**
- **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**
- **Radius**
- **Central angle**: Zwischen 0° und 360°
- **Semi-axis a**
- **Semi-axis b**
- **Outer radius (R)**
- **Inner radius (r)**
- **Ergebniseinheit** (Optionen: Millimeter (mm), Zentimeter (cm), Meter (m), Kilometer (km), Zoll (in), Fuß (ft), Yard (yd), Meilen (mi))

## Ergebnisse

- Fläche — Hauptergebnis
- Umfang
- Diagonal
- Second diagonal
- Side length
- Höhe
- Bogenlänge
- Sehnenlänge
- Apothem (inradius)
- Circumradius
- Ring width

## Formel

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

## Durchgerechnete Beispiele

### Rectangle 8 × 5 cm

- Shape: Rectangle
- Länge: 8 cm
- Breite: 5 cm
- Ergebniseinheit: Zentimeter (cm)
- **Fläche: 40 cm²**
- **Umfang: 26 cm**
- **Diagonal: 9.433981 cm**
- Prüfquelle: 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
- Ergebniseinheit: Zentimeter (cm)
- **Fläche: 6 cm²**
- **Umfang: 12 cm**
- **Höhe: 4 cm**
- Prüfquelle: 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
- Ergebniseinheit: Meter (m)
- **Fläche: 10.392305 m²**
- **Umfang: 12 m**
- **Apothem (inradius): 1.732051 m**
- **Circumradius: 2 m**
- Prüfquelle: 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
- Ergebniseinheit: Zentimeter (cm)
- **Fläche: 9 cm²**
- **Apothem (inradius): 1.5 cm**
- **Circumradius: 2.12132 cm**
- Prüfquelle: 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
- Ergebniseinheit: Zentimeter (cm)
- **Fläche: 78.539816 cm²**
- **Umfang: 31.415927 cm**
- Prüfquelle: 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
- Ergebniseinheit: Zentimeter (cm)
- **Fläche: 188.495559 cm²**
- **Umfang: 51.053998 cm**
- Prüfquelle: 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

## Fragen

### 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.

### Wie genau arbeitet „Fläche und Umfang ebener Figuren berechnen“?

Die Genauigkeit hängt von Ihren Eingaben und den Annahmen der Methode ab. Die Dezimalrechnung nutzt 50 signifikante Stellen, doch Schätzungen, numerische Verfahren und Quelldaten können ungenauer sein. Die angezeigte Rundung beseitigt diese Grenzen nicht. Anhand unabhängiger Quellen geprüfte Rechenbeispiele: 16. Beispielsweise wird „Rectangle 8 × 5 cm“ anhand von 8 × 5, 2(8 + 5); Python 3.8 math.sqrt(89) geprüft.

### Woher stammt die Methode?

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).

## Quellen

- [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)