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.
A 5-sided polygon splits into 3 triangles from one vertex.
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°.
검증 출처: Angle sum of a triangle (Euclid I.32); a triangle has no diagonals
검증 출처: 3 × 180; 540/5; 360/5; 5·2/2
검증 출처: Python 3.8 fractions: 900/7, 360/7; 7·4/2
검증 출처: 10 × 180; 1800/12; 12·9/2; Python 3.8 math: 10*pi
(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.
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°.
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.
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.
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.
정확도는 입력값과 계산 방법의 가정에 따라 달라집니다. 십진 연산은 유효숫자 50자리를 사용하지만, 추정값·수치해석 방법·원본 데이터의 정밀도는 더 낮을 수 있습니다. 표시값을 반올림해도 이러한 한계는 사라지지 않습니다. 독립적인 출처의 풀이와 대조한 계산 예시: 8. 예를 들어 “Triangle (edge case: no diagonals)”은 Angle sum of a triangle (Euclid I.32); a triangle has no diagonals와 대조해 확인합니다.
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).
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