# مسطح اشکال کا رقبہ اور محیط کیلکولیٹر

> مستطیل، مثلث، ذوزنقہ، دائرہ، بیضہ، باقاعدہ کثیرالاضلاع اور حلقے سمیت 15 اشکال کا رقبہ، محیط اور پیمانے کے مطابق خاکے دیکھیں۔

انٹرایکٹو صورت: https://www.calcopenly.com/ur/geometry/shape-area-perimeter-calculator
موضوع: ہندسی کیلکولیٹر

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.

## اندراجات

- **Shape** (اختیارات: Square, Rectangle, Triangle (base and height), Triangle (three sides), Parallelogram, Trapezoid (trapezium), Rhombus, Kite, دائرہ, Semicircle, Circular sector, Circular segment, Ellipse, Regular polygon, Annulus (ring))
- **Side length**
- **Number of sides**
- **لمبائی**
- **چوڑائی**
- **Base**
- **Slanted side**
- **Trapezoid defined by** (اختیارات: Bases and height, Bases and legs)
- **Bottom base (a)**
- **Top base (b)**
- **اونچائی**
- **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 قطر**
- **Central angle**: 0° اور 360° کے درمیان
- **Semi-axis a**
- **Semi-axis b**
- **Outer radius (R)**
- **Inner radius (r)**
- **نتائج کی اکائی** (اختیارات: ملی میٹر (⁨mm⁩), سینٹی میٹر (⁨cm⁩), میٹر (⁨m⁩), کلومیٹر (⁨km⁩), انچ (⁨in⁩), فٹ (⁨ft⁩), گز (⁨yd⁩), میل (⁨mi⁩))

## نتائج

- رقبہ — بنیادی نتیجہ
- محیط
- Diagonal
- Second diagonal
- Side length
- اونچائی
- قوس کی لمبائی
- وتر کی لمبائی
- Apothem (inradius)
- Circumradius
- Ring width

## فارمولا

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

## حل شدہ مثالیں

### Rectangle 8 × 5 cm

- Shape: Rectangle
- لمبائی: 8 cm
- چوڑائی: 5 cm
- نتائج کی اکائی: سینٹی میٹر (⁨cm⁩)
- **رقبہ: 40 cm²**
- **محیط: 26 cm**
- **Diagonal: 9.433981 cm**
- جانچ کا ماخذ: ⁨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
- نتائج کی اکائی: سینٹی میٹر (⁨cm⁩)
- **رقبہ: 6 cm²**
- **محیط: 12 cm**
- **اونچائی: 4 cm**
- جانچ کا ماخذ: ⁨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
- نتائج کی اکائی: میٹر (⁨m⁩)
- **رقبہ: 10.392305 m²**
- **محیط: 12 m**
- **Apothem (inradius): 1.732051 m**
- **Circumradius: 2 m**
- جانچ کا ماخذ: ⁨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
- نتائج کی اکائی: سینٹی میٹر (⁨cm⁩)
- **رقبہ: 9 cm²**
- **Apothem (inradius): 1.5 cm**
- **Circumradius: 2.12132 cm**
- جانچ کا ماخذ: ⁨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
- نتائج کی اکائی: سینٹی میٹر (⁨cm⁩)
- **رقبہ: 78.539816 cm²**
- **محیط: 31.415927 cm**
- جانچ کا ماخذ: ⁨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
- نتائج کی اکائی: سینٹی میٹر (⁨cm⁩)
- **رقبہ: 188.495559 cm²**
- **محیط: 51.053998 cm**
- جانچ کا ماخذ: ⁨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

## سوالات

### 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.

### “⁨مسطح اشکال کا رقبہ اور محیط کیلکولیٹر⁩” کتنا درست ہے؟

درستی آپ کی درج کردہ قدروں اور طریقے کے مفروضوں پر منحصر ہے۔ اعشاری حساب 50 بامعنی ہندسے استعمال کرتا ہے، مگر تخمینے، عددی طریقے اور ماخذ کا ڈیٹا کم درست ہو سکتے ہیں؛ دکھائی گئی قدروں کو راؤنڈ کرنے سے یہ حدود ختم نہیں ہوتیں۔ آزاد ذرائع کی حل شدہ مثالوں سے جانچ: 16۔ مثلاً، “⁨Rectangle 8 × 5 cm⁩” کو ⁨8 × 5, 2(8 + 5); Python 3.8 math.sqrt(89)⁩ سے جانچا جاتا ہے۔

### اس طریقے کا ماخذ کیا ہے؟

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).

## ماخذ

- [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)