# حاسبة مساحة الأشكال المستوية ومحيطها

> احسب مساحة 15 شكلًا ومحيطها، مثل المستطيلات والمثلثات وشبه المنحرف والدوائر والأشكال البيضاوية والمضلعات المنتظمة والحلقات، مع رسم بمقياس.

النسخة التفاعلية: https://www.calcopenly.com/ar/geometry/shape-area-perimeter-calculator
الموضوع: حاسبات الهندسة

Choose one of 15 plane shapes and enter the dimensions that define it. The area and perimeter come back with the other lengths that shape has: diagonals, heights, arc and chord lengths, a regular polygon's apothem and circumradius, or a ring's width. A triangle from three sides uses Heron's formula, A = √(s(s − a)(s − b)(s − c)) with s half the perimeter, and a regular polygon uses A = ns²/(4 tan(180°/n)).

The default rectangle, 8 cm by 5 cm, has an area of 40 cm², a perimeter of 26 cm and a diagonal of 9.434 cm. Area sizes flooring, turf and paint; perimeter sizes fencing, edging and skirting board. The drawing is to scale, so a mistyped dimension shows up as the wrong shape.

Base and height do not fix a triangle's slanted sides, and a trapezoid needs its legs, so those two inputs give area only. An ellipse's perimeter has no closed form; Ramanujan's second approximation is used, accurate to 0.04% or better.

## المدخلات

- **Shape** (الخيارات: Square, Rectangle, Triangle (base and height), Triangle (three sides), Parallelogram, Trapezoid (trapezium), Rhombus, Kite, الدائرة, Semicircle, Circular sector, Circular segment, Ellipse, Regular polygon, Annulus (ring))
- **Side length**
- **Number of sides**
- **الطول**
- **العرض**
- **Base**
- **Slanted side**
- **Trapezoid defined by** (الخيارات: Bases and height, Bases and legs)
- **Bottom base (a)**
- **Top base (b)**
- **الارتفاع**
- **Left leg (c)**
- **Right leg (d)**
- **Side a**
- **Side b**
- **Side c**
- **Diagonal p**
- **Diagonal q**
- **Shorter pair of sides (a)**
- **Longer pair of sides (b)**
- **Angle between a and b**
- **نصف القطر**
- **Central angle**: بين 0° و360°
- **Semi-axis a**
- **Semi-axis b**
- **Outer radius (R)**
- **Inner radius (r)**
- **وحدة النتائج** (الخيارات: مليمتر (⁨mm⁩), سنتيمتر (⁨cm⁩), متر (⁨m⁩), كيلومتر (⁨km⁩), بوصة (⁨in⁩), قدم (⁨ft⁩), ياردة (⁨yd⁩), ميل (⁨mi⁩))

## النتائج

- المساحة — النتيجة الرئيسية
- المحيط
- Diagonal
- Second diagonal
- Side length
- الارتفاع
- طول القوس
- طول الوتر
- Apothem (inradius)
- Circumradius
- Ring width

## الصيغة

$$
\begin{gathered} A_\triangle = \sqrt{s(s-a)(s-b)(s-c)},\quad A_{n\text{-gon}} = \frac{n s^2}{4\tan(\pi/n)} \\ P_\text{ellipse} \approx \pi(a+b)\left(1 + \frac{3h}{10 + \sqrt{4-3h}}\right) \end{gathered}
$$

## أمثلة محلولة

### Rectangle 8 × 5 cm

- Shape: Rectangle
- الطول: 8 cm
- العرض: 5 cm
- وحدة النتائج: سنتيمتر (⁨cm⁩)
- **المساحة: 40 cm²**
- **المحيط: 26 cm**
- **Diagonal: 9.433981 cm**
- مصدر التحقق: ⁨8 × 5, 2(8 + 5); Python 3.8 math.sqrt(89)⁩

### 3-4-5 triangle by Heron's formula

- Shape: Triangle (three sides)
- Side a: 3 cm
- Side b: 4 cm
- Side c: 5 cm
- وحدة النتائج: سنتيمتر (⁨cm⁩)
- **المساحة: 6 cm²**
- **المحيط: 12 cm**
- **الارتفاع: 4 cm**
- مصدر التحقق: ⁨Right triangle with legs 3 and 4: ½·3·4 = 6; height onto side 3 is the other leg, 4⁩

### Regular hexagon, side 2 m

- Shape: Regular polygon
- Side length: 2 m
- Number of sides: 6
- وحدة النتائج: متر (⁨m⁩)
- **المساحة: 10.392305 m²**
- **المحيط: 12 m**
- **Apothem (inradius): 1.732051 m**
- **Circumradius: 2 m**
- مصدر التحقق: ⁨Python 3.8 math: 6*4/(4*tan(pi/6)) = 6√3, 2/(2*tan(pi/6)) = √3; a hexagon's circumradius equals its side⁩

### Square as a regular 4-gon (exact tan 45°)

- Shape: Regular polygon
- Side length: 3 cm
- Number of sides: 4
- وحدة النتائج: سنتيمتر (⁨cm⁩)
- **المساحة: 9 cm²**
- **Apothem (inradius): 1.5 cm**
- **Circumradius: 2.12132 cm**
- مصدر التحقق: ⁨Square of side 3: area 9, apothem 3/2; Python 3.8 math: 3/sqrt(2)⁩

### Ellipse with a = b is a circle (edge case)

- Shape: Ellipse
- Semi-axis a: 5 cm
- Semi-axis b: 5 cm
- وحدة النتائج: سنتيمتر (⁨cm⁩)
- **المساحة: 78.539816 cm²**
- **المحيط: 31.415927 cm**
- مصدر التحقق: ⁨h = 0 so Ramanujan II reduces to 2πr; Python 3.8 math: 25*pi, 10*pi⁩

### Ellipse 10 × 6 (semi-axes)

- Shape: Ellipse
- Semi-axis a: 10 cm
- Semi-axis b: 6 cm
- وحدة النتائج: سنتيمتر (⁨cm⁩)
- **المساحة: 188.495559 cm²**
- **المحيط: 51.053998 cm**
- مصدر التحقق: ⁨Python 3.8 math: 60*pi and Ramanujan II with h = 1/16; the exact perimeter by the AGM series (Python decimal, 40 digits) is 51.0539977268, 1.2e-9 higher⁩

## الأسئلة

### How do you find the area of a trapezoid?

Average the two parallel sides and multiply by the perpendicular height: A = (a + b)/2 × h. Bases of 10 cm and 4 cm with a height of 4 cm give 7 × 4 = 28 cm². British English calls this shape a trapezium; the formula is the same. The perimeter also needs the two slanted legs: legs of 5 cm each make it 24 cm.

### How do you find the area of a triangle from three sides?

Use Heron's formula. Take the semi-perimeter s = (a + b + c)/2, then A = √(s(s − a)(s − b)(s − c)). Sides of 5, 6 and 7 cm give s = 9 and A = √(9 × 4 × 3 × 2) = √216 ≈ 14.697 cm². Three lengths only make a triangle if each is shorter than the other two combined, so 3, 4 and 8 have no area.

### What is the area of a regular hexagon?

A = (3√3/2)s², about 2.598 times the side squared. A hexagon with 5 cm sides covers 64.952 cm², and one with 2 m sides covers 10.392 m². Any regular polygon with n sides of length s has A = ns²/(4 tan(180°/n)). A hexagon's circumradius equals its side, and its apothem is s√3/2 ≈ 0.866s.

### Is there a formula for the perimeter of an ellipse?

Not an exact one in elementary functions: the exact perimeter is a complete elliptic integral. Ramanujan's 1914 approximation, P ≈ π(a + b)(1 + 3h/(10 + √(4 − 3h))) with h = (a − b)²/(a + b)², is off by at most 0.04%, reached only as the ellipse flattens into a line. For semi-axes of 10 and 6 it gives 51.0539977, within 1.2 × 10⁻⁹ of the exact value.

### How do you convert square feet to square metres?

Multiply by 0.09290304, the square of the international foot (0.3048 m exactly, fixed in 1959). One square metre is about 10.764 ft². A 12 ft × 10 ft room is 120 ft², or 11.148 m². Area scales with the square of the length unit, so converting each side first and then multiplying gives the same result.

### ما مدى دقة «⁨حاسبة مساحة الأشكال المستوية ومحيطها⁩»؟

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: 16. مثلًا، يجري التحقق من «⁨Rectangle 8 × 5 cm⁩» بالرجوع إلى ⁨8 × 5, 2(8 + 5); Python 3.8 math.sqrt(89)⁩.

### ما مصدر هذه الطريقة؟

Weisstein, E. W. “Heron's Formula”, “Regular Polygon”, “Ellipse” — MathWorld; Ramanujan, S. (1914) “Modular equations and approximations to π”, Quarterly Journal of Mathematics 45, 350–372 (perimeter approximation II); NIST Handbook 44, Appendix C — exact inch–centimetre relation (1 in = 2.54 cm).

## المصادر

- [Weisstein, E. W. “Heron's Formula”, “Regular Polygon”, “Ellipse” — MathWorld](https://mathworld.wolfram.com/Ellipse.html)
- Ramanujan, S. (1914) “Modular equations and approximations to π”, Quarterly Journal of Mathematics 45, 350–372 (perimeter approximation II)
- [NIST Handbook 44, Appendix C — exact inch–centimetre relation (1 in = 2.54 cm)](https://www.nist.gov/pml/owm/nist-handbook-44-current-edition)