Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, b and angle β.

Obtuse scalene triangle.

Sides: a = 1.27   b = 3.78   c = 4.5366427824

Area: T = 2.09661349022
Perimeter: p = 9.5866427824
Semiperimeter: s = 4.7933213912

Angle ∠ A = α = 14.15111081903° = 14°9'4″ = 0.24769834307 rad
Angle ∠ B = β = 46.691° = 46°41'28″ = 0.8154911681 rad
Angle ∠ C = γ = 119.158789181° = 119°9'28″ = 2.08796975418 rad

Height: ha = 3.30109998459
Height: hb = 1.10990660858
Height: hc = 0.92441345761

Median: ma = 4.12768103544
Median: mb = 2.74329798944
Median: mc = 1.67550688492

Inradius: r = 0.43773130306
Circumradius: R = 2.59773489814

Vertex coordinates: A[4.5366427824; 0] B[0; 0] C[0.87111344817; 0.92441345761]
Centroid: CG[1.80325207686; 0.30880448587]
Coordinates of the circumscribed circle: U[2.2688213912; -1.265547516]
Coordinates of the inscribed circle: I[1.0133213912; 0.43773130306]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.849889181° = 165°50'56″ = 0.24769834307 rad
∠ B' = β' = 133.309° = 133°18'32″ = 0.8154911681 rad
∠ C' = γ' = 60.84221081903° = 60°50'32″ = 2.08796975418 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and angle β.

a = 1.27 ; ; b = 3.78 ; ; beta = 46.691° ; ;

2. From angle β, side a and b we calculate c - by using the law of cosines and quadratic equation:

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 3.78**2 = 1.27**2 + c**2 - 2 * 1.27 * c * cos(46° 41'28") ; ; ; ; ; ; c**2 -1.742c -12.676 =0 ; ; a=1; b=-1.742; c=-12.676 ; ; D = b**2 - 4ac = 1.742**2 - 4 * 1 * (-12.676) = 53.7375011407 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 1.74 ± sqrt{ 53.74 } }{ 2 } ; ; c_{1,2} = 0.87113448 ± 3.66529334231 ; ; c_{1} = 4.53642782231 ; ; c_{2} = -2.79415886231 ; ;
 ; ; text{ Factored form: } ; ; (c -4.53642782231) (c +2.79415886231) = 0 ; ; ; ; c > 0 ; ; ; ; c = 4.536 ; ;


Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 1.27 ; ; b = 3.78 ; ; c = 4.54 ; ;

3. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 1.27+3.78+4.54 = 9.59 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 9.59 }{ 2 } = 4.79 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4.79 * (4.79-1.27)(4.79-3.78)(4.79-4.54) } ; ; T = sqrt{ 4.39 } = 2.1 ; ;

6. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.1 }{ 1.27 } = 3.3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.1 }{ 3.78 } = 1.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.1 }{ 4.54 } = 0.92 ; ;

7. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 1.27**2-3.78**2-4.54**2 }{ 2 * 3.78 * 4.54 } ) = 14° 9'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.78**2-1.27**2-4.54**2 }{ 2 * 1.27 * 4.54 } ) = 46° 41'28" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.54**2-1.27**2-3.78**2 }{ 2 * 3.78 * 1.27 } ) = 119° 9'28" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.1 }{ 4.79 } = 0.44 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1.27 }{ 2 * sin 14° 9'4" } = 2.6 ; ;

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