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Çokgen açıları: iç ve dış açılar

Kenar sayısından veya bir açıdan düzgün çokgenin iç açıları toplamını, her iç ve dış açısını ve köşegen sayısını bulun.

Güncellendi Doğrulanan örnekler: 8

Dene
Sum of interior angles
°
Sum of interior angles: 540 °
En fazla ondalık basamak: 4; En yakına; eşit uzaklıkta sıfırdan uzağa
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°
Nasıl hesaplanır 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

Çokgen açıları: iç ve dış açılar hakkında

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°.

Çözümlü örnekler

Triangle (edge case: no diagonals)

Bilinen ölçü
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

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

Pentagon

Bilinen ölçü
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

Doğrulama kaynağı: 3 × 180; 540/5; 360/5; 5·2/2

Heptagon (repeating decimal)

Bilinen ölçü
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

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

Dodecagon

Bilinen ölçü
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

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

Sorular

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.

“Çokgen açıları: iç ve dış açılar” ne kadar doğru sonuç verir?

Doğruluk, girdilerinize ve yöntemin varsayımlarına bağlıdır. Ondalık aritmetik 50 anlamlı basamak kullanır; ancak tahminler, sayısal yöntemler ve kaynak veriler daha az hassas olabilir. Gösterilen değerin yuvarlanması bu sınırları ortadan kaldırmaz. Bağımsız kaynaklarla doğrulanan çözümlü örnek sayısı: 8. Örneğin “Triangle (edge case: no diagonals)”, Angle sum of a triangle (Euclid I.32); a triangle has no diagonals ile karşılaştırılarak doğrulanır.

Yöntemin kaynağı nedir?

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).

Bu hesaplayıcı hakkında

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}

Kaynaklar

  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)

Kaynaklarla doğrulandı

Bu hesaplayıcı, yanıtları bağımsız kaynaklardan alınan 8 çözümlü örnek içerir. Bunlar test paketinde çalıştırılır; burada da çalıştırabilirsiniz.

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