# قائمہ مثلث کیلکولیٹر: وتر، اضلاع اور زاویے > دو قدروں سے فیثاغورث کے قضیے اور مثلثیات کی مدد سے وتر، نامعلوم ضلع، حاد زاویے، رقبہ اور محیط نکالیں۔ انٹرایکٹو صورت: https://www.calcopenly.com/ur/geometry/right-triangle-calculator موضوع: ہندسی کیلکولیٹر 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°. ## اندراجات - **I know** (اختیارات: 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° - **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 - Leg a - Leg b - Hypotenuse c - Angle α (°) - Angle β (°) - محیط - Height onto the hypotenuse - Inradius - Circumradius ## فارمولا $$ a^2 + b^2 = c^2,\quad \sin\alpha = \frac{a}{c},\quad \cos\alpha = \frac{b}{c},\quad \tan\alpha = \frac{a}{b},\quad \alpha + \beta = 90^\circ $$ ## حل شدہ مثالیں ### Legs 3 and 4 - I know: Both legs (a and b) - Leg a (opposite α): 3 cm - Leg b (opposite β): 4 cm - نتائج کی اکائی: سینٹی میٹر (⁨cm⁩) - **Hypotenuse c: 5 cm** - **رقبہ: 6 cm²** - **Angle α: 36.869898 °** - **Angle β: 53.130102 °** - **Height onto the hypotenuse: 2.4 cm** - **Inradius: 1 cm** - **Circumradius: 2.5 cm** - **محیط: 12 cm** - جانچ کا ماخذ: ⁨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 - نتائج کی اکائی: سینٹی میٹر (⁨cm⁩) - **Leg b: 12 cm** - **رقبہ: 30 cm²** - **محیط: 30 cm** - **Angle α: 22.619865 °** - جانچ کا ماخذ: ⁨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 ° - نتائج کی اکائی: سینٹی میٹر (⁨cm⁩) - **Leg a: 5 cm** - **Leg b: 8.660254 cm** - **Angle β: 60 °** - **رقبہ: 21.650635 cm²** - جانچ کا ماخذ: ⁨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 ° - نتائج کی اکائی: سینٹی میٹر (⁨cm⁩) - **Leg b: 7 cm** - **Hypotenuse c: 9.899495 cm** - **رقبہ: 24.5 cm²** - **Angle β: 45 °** - جانچ کا ماخذ: ⁨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 ° - نتائج کی اکائی: سینٹی میٹر (⁨cm⁩) - **Leg a: 5.196152 cm** - **Hypotenuse c: 10.392305 cm** - **Angle α: 30 °** - جانچ کا ماخذ: ⁨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 - نتائج کی اکائی: سینٹی میٹر (⁨cm⁩) - **Hypotenuse c: 25.4 cm** - **رقبہ: 154.8384 cm²** - جانچ کا ماخذ: ⁨6-8-10 in triangle; 10 in = 25.4 cm and 24 in² = 24 × 6.4516 cm² exactly (1 in = 2.54 cm)⁩ ## سوالات ### 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. ### “⁨قائمہ مثلث کیلکولیٹر: وتر، اضلاع اور زاویے⁩” 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: 6۔ مثلاً، “⁨Legs 3 and 4⁩” کو ⁨3-4-5 triple; h = 3·4/5; r = (3 + 4 − 5)/2; Python 3.8 math: degrees(atan(3/4))⁩ سے جانچا جاتا ہے۔ ### اس طریقے کا ماخذ کیا ہے؟ Euclid, Elements, Book I, Proposition 47 (Pythagorean theorem); OpenStax Precalculus 2e, §5.4 Right Triangle Trigonometry. ## ماخذ - [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)