# 多角形の角度計算：内角・外角

> 辺の数または1つの角度から、正多角形の内角の和、各内角と外角、対角線の数を計算します。

操作できるページ：https://www.calcopenly.com/ja/geometry/polygon-angles-calculator
分野：幾何の計算機

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°.

## 入力

- **既知の値** (選択肢：Number of sides, Interior angle, Exterior angle)
- **Number of sides**
- **Each interior angle**
- **Each exterior angle**

## 結果

- Sum of interior angles (°) — 主な結果
- Each interior angle (regular polygon) (°)
- Each exterior angle (regular polygon) (°)
- Sum of exterior angles (°)
- Central angle (regular polygon) (°)
- Number of diagonals
- Triangles from one vertex
- Sum of interior angles in radians (rad)
- Number of sides
- Name

## 計算式

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

## 計算例

### Triangle (edge case: no diagonals)

- 既知の値: 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**
- 照合元：Angle sum of a triangle (Euclid I.32); a triangle has no diagonals

### Pentagon

- 既知の値: 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**
- 照合元：3 × 180; 540/5; 360/5; 5·2/2

### Heptagon (repeating decimal)

- 既知の値: 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**
- 照合元：Python 3.8 fractions: 900/7, 360/7; 7·4/2

### Dodecagon

- 既知の値: 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**
- 照合元：10 × 180; 1800/12; 12·9/2; Python 3.8 math: 10*pi

### Interior angle 140°

- 既知の値: Interior angle
- Each interior angle: 140 °
- **Number of sides: 9**
- **Name: Nonagon**
- **Number of diagonals: 27**
- **Sum of interior angles: 1,260 °**
- 照合元：360/(180 − 140) = 9; 9·6/2; 7 × 180

### Exterior angle 24°

- 既知の値: Exterior angle
- Each exterior angle: 24 °
- **Number of sides: 15**
- **Sum of interior angles: 2,340 °**
- **Each interior angle (regular polygon): 156 °**
- 照合元：360/24 = 15; 13 × 180; 180 − 24

## よくある質問

### 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.

### 「多角形の角度計算：内角・外角」の精度はどのくらいですか？

精度は入力値と計算方法の前提に依存します。十進演算には有効数字50桁を使いますが、推定、数値計算手法、元データの精度はそれより低い場合があります。表示の丸め処理でこれらの制約がなくなるわけではありません。 独立した出典の解答と照合した計算例：8。 例えば、「Triangle (edge case: no diagonals)」はAngle sum of a triangle (Euclid I.32); a triangle has no diagonalsと照合しています。

### この計算方法の出典は何ですか？

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).

## 出典

- [Weisstein, E. W. “Polygon”, “Regular Polygon”, “Polygon Diagonal” — MathWorld](https://mathworld.wolfram.com/PolygonDiagonal.html)
- [Euclid, Elements, Book I, Proposition 32 (angle sum of a triangle, extended to polygons by triangulation)](https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI32.html)