Triangle calculator

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

You have entered side b, c and angle γ.

Acute scalene triangle.

Sides: a = 37.24552866818   b = 25.3   c = 31.2

Area: T = 390.6033437057
Perimeter: p = 93.74552866818
Semiperimeter: s = 46.87326433409

Angle ∠ A = α = 81.75879229994° = 81°45'29″ = 1.42769449459 rad
Angle ∠ B = β = 42.24220770006° = 42°14'31″ = 0.73772633265 rad
Angle ∠ C = γ = 56° = 0.97773843811 rad

Height: ha = 20.97546505856
Height: hb = 30.87877420598
Height: hc = 25.03986818626

Median: ma = 21.44767283052
Median: mb = 31.94221851163
Median: mc = 27.75441112271

Inradius: r = 8.33332922834
Circumradius: R = 18.81769999967

Vertex coordinates: A[31.2; 0] B[0; 0] C[27.57330990386; 25.03986818626]
Centroid: CG[19.59110330129; 8.34662272875]
Coordinates of the circumscribed circle: U[15.6; 10.52223328627]
Coordinates of the inscribed circle: I[21.57326433409; 8.33332922834]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 98.24220770006° = 98°14'31″ = 1.42769449459 rad
∠ B' = β' = 137.7587922999° = 137°45'29″ = 0.73772633265 rad
∠ C' = γ' = 124° = 0.97773843811 rad

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How did we calculate this triangle?

1. Input data entered: side b, c and angle γ.

b = 25.3 ; ; c = 31.2 ; ; gamma = 56° ; ;

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

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 31.2**2 = 25.3**2 + a**2 - 2 * 25.3 * a * cos(56° ) ; ; ; ; ; ; a**2 -28.295a -333.35 =0 ; ; a=1; b=-28.295; c=-333.35 ; ; D = b**2 - 4ac = 28.295**2 - 4 * 1 * (-333.35) = 2134.01613124 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 28.3 ± sqrt{ 2134.02 } }{ 2 } ; ; a_{1,2} = 14.14758046 ± 23.097706224 ; ; a_{1} = 37.245286684 ; ; a_{2} = -8.950125764 ; ;
 ; ; (a -37.245286684) (a +8.950125764) = 0 ; ; ; ; a > 0 ; ; ; ; a = 37.245 ; ;


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 = 37.25 ; ; b = 25.3 ; ; c = 31.2 ; ;

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

p = a+b+c = 37.25+25.3+31.2 = 93.75 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 93.75 }{ 2 } = 46.87 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 46.87 * (46.87-37.25)(46.87-25.3)(46.87-31.2) } ; ; T = sqrt{ 152571.05 } = 390.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 390.6 }{ 37.25 } = 20.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 390.6 }{ 25.3 } = 30.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 390.6 }{ 31.2 } = 25.04 ; ;

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{ 37.25**2-25.3**2-31.2**2 }{ 2 * 25.3 * 31.2 } ) = 81° 45'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 25.3**2-37.25**2-31.2**2 }{ 2 * 37.25 * 31.2 } ) = 42° 14'31" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 31.2**2-37.25**2-25.3**2 }{ 2 * 25.3 * 37.25 } ) = 56° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 390.6 }{ 46.87 } = 8.33 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37.25 }{ 2 * sin 81° 45'29" } = 18.82 ; ;




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