# حاسبة المثلث القائم: الوتر والأضلاع والزوايا

> احسب الوتر والضلع المجهول والزاويتين الحادتين والمساحة والمحيط من قيمتين باستخدام نظرية فيثاغورس والنسب المثلثية.

النسخة التفاعلية: https://www.calcopenly.com/ar/geometry/right-triangle-calculator
الموضوع: حاسبات الهندسة

A right triangle is fixed by any two of its parts as long as one of them is a side. With both legs known, Pythagoras' theorem gives the hypotenuse c = √(a² + b²). With a side and an acute angle, the rest follows from sin α = a/c, cos α = b/c and tan α = a/b, and the two acute angles always add up to 90°.

The default legs of 3 cm and 4 cm make the 3-4-5 triangle: hypotenuse 5 cm, area 6 cm², perimeter 12 cm and angles of 36.8699° and 53.1301°. Builders use that ratio to square a corner, since marks 3 m and 4 m from it along two walls should be exactly 5 m apart. The results also include the height onto the hypotenuse (ab/c), the inradius ((a + b − c)/2) and the circumradius (c/2).

Angle α is opposite leg a and β is opposite leg b. Angles can be typed in degrees, radians, gradians or turns and must lie strictly between 0° and 90°.

## المدخلات

- **I know** (الخيارات: Both legs (a and b), Leg a and hypotenuse c, Leg b and hypotenuse c, Leg a and angle α, Leg b and angle α, Hypotenuse c and angle α, Leg a and angle β, Leg b and angle β, Hypotenuse c and angle β)
- **Leg a (opposite α)**
- **Leg b (opposite β)**
- **Hypotenuse c**
- **Angle α**: Between 0° and 90°
- **Angle β**: Between 0° and 90°
- **وحدة النتائج** (الخيارات: مليمتر (⁨mm⁩), سنتيمتر (⁨cm⁩), متر (⁨m⁩), كيلومتر (⁨km⁩), بوصة (⁨in⁩), قدم (⁨ft⁩), ياردة (⁨yd⁩), ميل (⁨mi⁩))

## النتائج

- المساحة — النتيجة الرئيسية
- Leg a
- Leg b
- Hypotenuse c
- Angle α (°)
- Angle β (°)
- المحيط
- Height onto the hypotenuse
- Inradius
- Circumradius

## الصيغة

$$
a^2 + b^2 = c^2,\quad \sin\alpha = \frac{a}{c},\quad \cos\alpha = \frac{b}{c},\quad \tan\alpha = \frac{a}{b},\quad \alpha + \beta = 90^\circ
$$

## أمثلة محلولة

### Legs 3 and 4

- I know: Both legs (a and b)
- Leg a (opposite α): 3 cm
- Leg b (opposite β): 4 cm
- وحدة النتائج: سنتيمتر (⁨cm⁩)
- **Hypotenuse c: 5 cm**
- **المساحة: 6 cm²**
- **Angle α: 36.869898 °**
- **Angle β: 53.130102 °**
- **Height onto the hypotenuse: 2.4 cm**
- **Inradius: 1 cm**
- **Circumradius: 2.5 cm**
- **المحيط: 12 cm**
- مصدر التحقق: ⁨3-4-5 triple; h = 3·4/5; r = (3 + 4 − 5)/2; Python 3.8 math: degrees(atan(3/4))⁩

### Leg 5 and hypotenuse 13

- I know: Leg a and hypotenuse c
- Leg a (opposite α): 5 cm
- Hypotenuse c: 13 cm
- وحدة النتائج: سنتيمتر (⁨cm⁩)
- **Leg b: 12 cm**
- **المساحة: 30 cm²**
- **المحيط: 30 cm**
- **Angle α: 22.619865 °**
- مصدر التحقق: ⁨5-12-13 triple; Python 3.8 math: degrees(asin(5/13))⁩

### 30-60-90 from hypotenuse 10 and α = 30°

- I know: Hypotenuse c and angle α
- Hypotenuse c: 10 cm
- Angle α: 30 °
- وحدة النتائج: سنتيمتر (⁨cm⁩)
- **Leg a: 5 cm**
- **Leg b: 8.660254 cm**
- **Angle β: 60 °**
- **المساحة: 21.650635 cm²**
- مصدر التحقق: ⁨Side opposite 30° is half the hypotenuse; Python 3.8 math: 10*cos(radians(30)), 25*sqrt(3)/2⁩

### 45° angle gives equal legs (edge case)

- I know: Leg a and angle α
- Leg a (opposite α): 7 cm
- Angle α: 45 °
- وحدة النتائج: سنتيمتر (⁨cm⁩)
- **Leg b: 7 cm**
- **Hypotenuse c: 9.899495 cm**
- **المساحة: 24.5 cm²**
- **Angle β: 45 °**
- مصدر التحقق: ⁨tan 45° = 1 so b = a; Python 3.8 math: 7*sqrt(2)⁩

### Leg b = 9 and β = 60°

- I know: Leg b and angle β
- Leg b (opposite β): 9 cm
- Angle β: 60 °
- وحدة النتائج: سنتيمتر (⁨cm⁩)
- **Leg a: 5.196152 cm**
- **Hypotenuse c: 10.392305 cm**
- **Angle α: 30 °**
- مصدر التحقق: ⁨Python 3.8 math: 9/tan(radians(60)), 9/sin(radians(60))⁩

### Legs 6 in and 8 in, results in cm

- I know: Both legs (a and b)
- Leg a (opposite α): 6 in
- Leg b (opposite β): 8 in
- وحدة النتائج: سنتيمتر (⁨cm⁩)
- **Hypotenuse c: 25.4 cm**
- **المساحة: 154.8384 cm²**
- مصدر التحقق: ⁨6-8-10 in triangle; 10 in = 25.4 cm and 24 in² = 24 × 6.4516 cm² exactly (1 in = 2.54 cm)⁩

## الأسئلة

### How do you find the hypotenuse of a right triangle?

Square both legs, add them and take the square root: c = √(a² + b²). Legs of 3 and 4 give √25 = 5, and legs of 6 in and 8 in give 10 in. If you know one leg and the angle opposite it instead, divide by the sine: c = a / sin α, so a 5 cm leg opposite 30° means a 10 cm hypotenuse.

### How do you find a missing side of a right triangle?

For a missing leg, subtract the squares: b = √(c² − a²). A hypotenuse of 13 and a leg of 5 give √(169 − 25) = 12. The hypotenuse must be longer than either leg, or no right triangle exists. With one side and an angle, use SOHCAHTOA: opposite = hypotenuse × sin α and adjacent = hypotenuse × cos α.

### What are the side ratios of a 30-60-90 and a 45-45-90 triangle?

A 30-60-90 triangle has sides in the ratio 1 : √3 : 2, so the side opposite 30° is half the hypotenuse; a hypotenuse of 10 gives legs of 5 and 8.6603. A 45-45-90 triangle has equal legs and a hypotenuse √2 ≈ 1.4142 times a leg, so legs of 7 give a hypotenuse of 9.8995.

### What does SOHCAHTOA mean?

It is a memory aid for the three trigonometric ratios in a right triangle: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent. In the 3-4-5 triangle, the angle opposite the side of 3 has sin = 0.6, cos = 0.8 and tan = 0.75, and each inverse function returns the same 36.87°.

### How do you check a corner is square with the 3-4-5 rule?

Mark 3 units along one side and 4 along the other, measured from the corner; the diagonal between the marks is exactly 5 when the angle is 90°. A longer diagonal means the angle is too wide: 5.05 m on 3 m and 4 m legs is 91.2°, and 4.95 m is 88.8°. Multiples such as 6-8-10 or 9-12-15 give a more precise check on large slabs.

### ما مدى دقة «⁨حاسبة المثلث القائم: الوتر والأضلاع والزوايا⁩»؟

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: 6. مثلًا، يجري التحقق من «⁨Legs 3 and 4⁩» بالرجوع إلى ⁨3-4-5 triple; h = 3·4/5; r = (3 + 4 − 5)/2; Python 3.8 math: degrees(atan(3/4))⁩.

### ما مصدر هذه الطريقة؟

Euclid, Elements, Book I, Proposition 47 (Pythagorean theorem); OpenStax Precalculus 2e, §5.4 Right Triangle Trigonometry.

## المصادر

- [Euclid, Elements, Book I, Proposition 47 (Pythagorean theorem)](https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI47.html)
- [OpenStax Precalculus 2e, §5.4 Right Triangle Trigonometry](https://openstax.org/books/precalculus-2e/pages/5-4-right-triangle-trigonometry)