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Vieleckwinkel berechnen: Innen- und Außenwinkel

Aus Seitenzahl oder einem Winkel eines regelmäßigen Vielecks die Innenwinkelsumme, jeden Innen- und Außenwinkel sowie die Anzahl der Diagonalen berechnen.

Aktualisiert Geprüfte Beispiele: 8

Ausprobieren
Sum of interior angles
°
Sum of interior angles: 540 °
Maximale Nachkommastellen: 4; Zum nächsten Wert, bei Gleichstand von null weg
Each interior angle (regular polygon)
108°
Each exterior angle (regular polygon)
72°
Sum of exterior angles
360°
Central angle (regular polygon)
72°
Number of diagonals
5
Triangles from one vertex
3
Sum of interior angles in radians
9.42477796rad
Name
Pentagon

A pentagon (5 sides) has interior angles adding to 540°; when regular, each is 108° and each exterior angle is 72°. It has 5 diagonals.

Regular pentagon with diagonals from one vertex

108°72°
So wird gerechnet S

  1. Sum of interior angles

    S=(n−2)×180∘=3×180∘=540∘S = (n - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ

    A 5-sided polygon splits into 3 triangles from one vertex.

  2. Each angle of the regular polygon

    Sn=540∘5=108∘,360∘n=72∘\frac{S}{n} = \frac{540^\circ}{5} = 108^\circ,\qquad \frac{360^\circ}{n} = 72^\circ

  3. Diagonals

    n(n−3)2=5×22=5\frac{n(n - 3)}{2} = \frac{5 \times 2}{2} = 5

Über Vieleckwinkel berechnen: Innen- und Außenwinkel

From one vertex, a polygon with n sides splits into n − 2 triangles, so its interior angles add up to (n − 2) × 180°. In a regular polygon all the angles are equal: each interior angle is (n − 2) × 180°/n and each exterior angle 360°/n, because the exterior angles of a convex polygon make one full turn. The number of diagonals is n(n − 3)/2.

The default pentagon has interior angles adding up to 540°, or 108° each when regular, with 72° exterior angles and 5 diagonals. Working backwards is common in tiling and woodwork: an interior angle of 140° means 360/(180 − 140) = 9 sides, and each joint of a regular n-sided frame is mitred at 180°/n, 36° for a pentagon.

An angle that fits no whole number of sides, such as 100°, is rejected with the nearest valid polygon. A rounded entry is accepted when it matches an exact angle: 128.57° is read as the heptagon's 900/7°.

Durchgerechnete Beispiele

Triangle (edge case: no diagonals)

Bekannte Größe
Number of sides
Number of sides
3
Sum of interior angles
180 °
Each interior angle (regular polygon)
60 °
Each exterior angle (regular polygon)
120 °
Number of diagonals
0
Triangles from one vertex
1
Name
Triangle

Prüfquelle: Angle sum of a triangle (Euclid I.32); a triangle has no diagonals

Pentagon

Bekannte Größe
Number of sides
Number of sides
5
Sum of interior angles
540 °
Each interior angle (regular polygon)
108 °
Each exterior angle (regular polygon)
72 °
Number of diagonals
5

Prüfquelle: 3 × 180; 540/5; 360/5; 5·2/2

Heptagon (repeating decimal)

Bekannte Größe
Number of sides
Number of sides
7
Sum of interior angles
900 °
Each interior angle (regular polygon)
128.57142857 °
Each exterior angle (regular polygon)
51.42857143 °
Number of diagonals
14

Prüfquelle: Python 3.8 fractions: 900/7, 360/7; 7·4/2

Dodecagon

Bekannte Größe
Number of sides
Number of sides
12
Sum of interior angles
1,800 °
Each interior angle (regular polygon)
150 °
Number of diagonals
54
Name
Dodecagon
Sum of interior angles in radians
31.41592654 rad

Prüfquelle: 10 × 180; 1800/12; 12·9/2; Python 3.8 math: 10*pi

Fragen

What is the sum of the interior angles of a polygon?

(n − 2) × 180°, where n is the number of sides. A triangle has 180°, a quadrilateral 360°, a pentagon 540°, a hexagon 720° and an octagon 1,080°. The rule holds for any simple polygon, regular or not, because diagonals from one vertex split it into n − 2 triangles of 180° each.

How do you find each interior angle of a regular polygon?

Divide the angle sum by the number of sides: (n − 2) × 180°/n, which is the same as 180° − 360°/n. A regular hexagon has 120° angles, an octagon 135° and a dodecagon 150°. The angle approaches 180° as n grows; a 1,000-sided polygon has angles of 179.64°.

How do you find the number of sides from an interior angle?

Subtract the angle from 180° to get the exterior angle, then divide 360° by it: n = 360°/(180° − interior angle). An interior angle of 140° gives 360/40 = 9 sides, a nonagon. If the result is not a whole number, no regular polygon has that angle: 100° gives 4.5.

What do the exterior angles of a polygon add up to?

360° for any convex polygon, whatever the number of sides. Walking once around the boundary, you turn through each exterior angle and finish facing your starting direction, which is one full turn. In a regular polygon each exterior angle is 360°/n: 72° for a pentagon, 60° for a hexagon and 45° for an octagon.

How many diagonals does a polygon have?

n(n − 3)/2. Each of the n vertices joins n − 3 others by a diagonal (not itself and not its two neighbours), and every diagonal is counted from both ends, hence the division by 2. A pentagon has 5 diagonals, a hexagon 9, an octagon 20 and a dodecagon 54; a triangle has none.

Wie genau arbeitet „Vieleckwinkel berechnen: Innen- und Außenwinkel“?

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: 8. Beispielsweise wird „Triangle (edge case: no diagonals)“ anhand von Angle sum of a triangle (Euclid I.32); a triangle has no diagonals geprüft.

Woher stammt die Methode?

Weisstein, E. W. “Polygon”, “Regular Polygon”, “Polygon Diagonal” — MathWorld; Euclid, Elements, Book I, Proposition 32 (angle sum of a triangle, extended to polygons by triangulation).

Über diesen Rechner

S=(n−2)×180∘,interior=(n−2)×180∘n,exterior=360∘n,diagonals=n(n−3)2S = (n - 2) \times 180^\circ,\quad \text{interior} = \frac{(n - 2) \times 180^\circ}{n},\quad \text{exterior} = \frac{360^\circ}{n},\quad \text{diagonals} = \frac{n(n - 3)}{2}

Quellen

  1. Weisstein, E. W. “Polygon”, “Regular Polygon”, “Polygon Diagonal” — MathWorld
  2. Euclid, Elements, Book I, Proposition 32 (angle sum of a triangle, extended to polygons by triangulation)

Anhand von Quellen geprüft

Dieser Rechner enthält 8 Rechenbeispiele mit Ergebnissen aus unabhängigen Quellen. Sie laufen in der Testsuite und können auch hier ausgeführt werden.

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