दोन ज्ञात मूल्यांवरून काटकोन त्रिकोणाचा कर्ण, अज्ञात बाजू, लघुकोन, क्षेत्रफळ आणि परिमिती पायथागोरस प्रमेय व त्रिकोणमितीने काढा.
अद्यतनित तपासलेली उदाहरणे: ६
Legs of 3 and 4 cm meet at the right angle; the hypotenuse is 5 cm, the acute angles are 36.8699° and 53.1301°, and the triangle encloses 6 cm².
The circumcentre of a right triangle is the midpoint of its hypotenuse.
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°.
पडताळणीचा स्रोत: 3-4-5 triple; h = 3·4/5; r = (3 + 4 − 5)/2; Python 3.8 math: degrees(atan(3/4))
पडताळणीचा स्रोत: 5-12-13 triple; Python 3.8 math: degrees(asin(5/13))
पडताळणीचा स्रोत: Side opposite 30° is half the hypotenuse; Python 3.8 math: 10*cos(radians(30)), 25*sqrt(3)/2
पडताळणीचा स्रोत: tan 45° = 1 so b = a; Python 3.8 math: 7*sqrt(2)
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.
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.
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.
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°.
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.
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.
Euclid, Elements, Book I, Proposition 47 (Pythagorean theorem); OpenStax Precalculus 2e, §5.4 Right Triangle Trigonometry.
या कॅल्क्युलेटरमध्ये स्वतंत्र स्रोतांमधील उत्तरांसह ६ सोडवलेली उदाहरणे आहेत. ती चाचणी संचात चालतात आणि तुम्ही ती येथेही चालवू शकता.
तीन मापांवरून त्रिकोणाच्या बाजू, कोन, क्षेत्रफळ, परिमिती, उंची, अंतर्वर्तुळ व परिवर्तुळ त्रिज्या काढा. SSA चे दोन्ही शक्य पर्याय पाहा.
अंश, रेडियन किंवा ग्रेडियनमध्ये sin, cos, tan, sec, csc व cot काढा. 15° च्या पटींना अचूक मूल्ये आणि व्यस्त फलनांची सर्व उत्तरे पाहा.
उभा चढ व आडवे अंतर यांवरून कोन, टक्केवारीतील उतार आणि छताचा कल (12 मध्ये x) काढा. तिरपी लांबी आणि राफ्टर गुणकही पाहा.