# काटकोन त्रिकोण कॅल्क्युलेटर: कर्ण, बाजू आणि कोन > दोन ज्ञात मूल्यांवरून काटकोन त्रिकोणाचा कर्ण, अज्ञात बाजू, लघुकोन, क्षेत्रफळ आणि परिमिती पायथागोरस प्रमेय व त्रिकोणमितीने काढा. परस्परसंवादी आवृत्ती: https://www.calcopenly.com/mr/geometry/right-triangle-calculator विषय: भूमितीचे कॅल्क्युलेटर 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°. ## इनपुट - **I know** (पर्याय: 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° - **निकालाचे एकक** (पर्याय: मिलीमीटर (mm), सेंटीमीटर (cm), मीटर (m), किलोमीटर (km), इंच (in), फूट (ft), यार्ड (yd), मैल (mi)) ## निकाल - क्षेत्रफळ — मुख्य निकाल - Leg a - Leg b - Hypotenuse c - Angle α (°) - Angle β (°) - परिमिती - Height onto the hypotenuse - Inradius - Circumradius ## सूत्र $$ a^2 + b^2 = c^2,\quad \sin\alpha = \frac{a}{c},\quad \cos\alpha = \frac{b}{c},\quad \tan\alpha = \frac{a}{b},\quad \alpha + \beta = 90^\circ $$ ## सोडवलेली उदाहरणे ### Legs 3 and 4 - I know: Both legs (a and b) - Leg a (opposite α): 3 cm - Leg b (opposite β): 4 cm - निकालाचे एकक: सेंटीमीटर (cm) - **Hypotenuse c: 5 cm** - **क्षेत्रफळ: 6 cm²** - **Angle α: 36.869898 °** - **Angle β: 53.130102 °** - **Height onto the hypotenuse: 2.4 cm** - **Inradius: 1 cm** - **Circumradius: 2.5 cm** - **परिमिती: 12 cm** - पडताळणीचा स्रोत: 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 - निकालाचे एकक: सेंटीमीटर (cm) - **Leg b: 12 cm** - **क्षेत्रफळ: 30 cm²** - **परिमिती: 30 cm** - **Angle α: 22.619865 °** - पडताळणीचा स्रोत: 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 ° - निकालाचे एकक: सेंटीमीटर (cm) - **Leg a: 5 cm** - **Leg b: 8.660254 cm** - **Angle β: 60 °** - **क्षेत्रफळ: 21.650635 cm²** - पडताळणीचा स्रोत: 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 ° - निकालाचे एकक: सेंटीमीटर (cm) - **Leg b: 7 cm** - **Hypotenuse c: 9.899495 cm** - **क्षेत्रफळ: 24.5 cm²** - **Angle β: 45 °** - पडताळणीचा स्रोत: 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 ° - निकालाचे एकक: सेंटीमीटर (cm) - **Leg a: 5.196152 cm** - **Hypotenuse c: 10.392305 cm** - **Angle α: 30 °** - पडताळणीचा स्रोत: 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 - निकालाचे एकक: सेंटीमीटर (cm) - **Hypotenuse c: 25.4 cm** - **क्षेत्रफळ: 154.8384 cm²** - पडताळणीचा स्रोत: 6-8-10 in triangle; 10 in = 25.4 cm and 24 in² = 24 × 6.4516 cm² exactly (1 in = 2.54 cm) ## प्रश्न ### 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. ### “काटकोन त्रिकोण कॅल्क्युलेटर: कर्ण, बाजू आणि कोन” किती अचूक आहे? अचूकता तुमच्या इनपुटवर आणि पद्धतीच्या गृहीतकांवर अवलंबून असते. दशांश गणना 50 सार्थ अंक वापरते, पण अंदाज, संख्यात्मक पद्धती आणि मूळ डेटा कमी अचूक असू शकतात; दाखवलेल्या मूल्यांचे पूर्णांकन केल्याने या मर्यादा दूर होत नाहीत. स्वतंत्र स्रोतांतील सोडवलेल्या उदाहरणांशी पडताळणी: ६. उदाहरणार्थ, “Legs 3 and 4” ची पडताळणी 3-4-5 triple; h = 3·4/5; r = (3 + 4 − 5)/2; Python 3.8 math: degrees(atan(3/4)) याच्याशी केली आहे. ### या पद्धतीचा स्रोत कोणता? Euclid, Elements, Book I, Proposition 47 (Pythagorean theorem); OpenStax Precalculus 2e, §5.4 Right Triangle Trigonometry. ## स्रोत - [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)