حاسبة زوايا المضلعات: الداخلية والخارجية

من عدد الأضلاع أو زاوية واحدة، احسب مجموع الزوايا الداخلية للمضلع المنتظم وكل زاوية داخلية وخارجية وعدد الأقطار.

آخر تحديث 28 سبتمبر 2026أمثلة تم التحقق منها: 8

Sum of interior angles

الحد الأقصى للمنازل العشرية: 4؛ إلى الأقرب، وعند التعادل بعيدًا عن الصفر

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

طريقة الحساب S

Sum of interior angles
$S = (n - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$
A 5-sided polygon splits into 3 triangles from one vertex.
Each angle of the regular polygon
$\frac{S}{n} = \frac{540^\circ}{5} = 108^\circ,\qquad \frac{360^\circ}{n} = 72^\circ$
Diagonals
$\frac{n(n - 3)}{2} = \frac{5 \times 2}{2} = 5$

حول حاسبة زوايا المضلعات: الداخلية والخارجية

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°.

أمثلة محلولة

Triangle (edge case: no diagonals)

القياس المعلوم: 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

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

Pentagon

القياس المعلوم: 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

مصدر التحقق: ⁨3 × 180; 540/5; 360/5; 5·2/2⁩

Heptagon (repeating decimal)

القياس المعلوم: 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

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

Dodecagon

القياس المعلوم: 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

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

الأسئلة

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.

ما مدى دقة «⁨حاسبة زوايا المضلعات: الداخلية والخارجية⁩»؟

تعتمد الدقة على مدخلاتك وافتراضات الطريقة. يستخدم الحساب العشري 50 رقمًا معنويًا، لكن التقديرات والأساليب العددية وبيانات المصدر قد تكون أقل دقة؛ تقريب القيم المعروضة لا يزيل هذه الحدود. أمثلة محلولة جرى التحقق منها بمصادر مستقلة: 8. مثلًا، يجري التحقق من «⁨Triangle (edge case: no diagonals)⁩» بالرجوع إلى ⁨Angle sum of a triangle (Euclid I.32); a triangle has no diagonals⁩.

ما مصدر هذه الطريقة؟

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).

حول هذه الحاسبة

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}

المصادر

تم التحقق بالرجوع إلى المصادر

تتضمن هذه الحاسبة أمثلة محلولة بإجابات من مصادر مستقلة، وعددها 8. تُشغّل ضمن مجموعة الاختبارات، ويمكنك تشغيلها هنا أيضًا.