Triangle calculator

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

You have entered side b, c and angle γ.

Obtuse scalene triangle.

Sides: a = 56.49439214689   b = 25.3   c = 31.2

Area: T = 11.64109167764
Perimeter: p = 112.9943921469
Semiperimeter: s = 56.49769607344

Angle ∠ A = α = 178.3109840523° = 178°18'35″ = 3.11220938058 rad
Angle ∠ B = β = 0.75768261438° = 0°45'25″ = 0.01332091081 rad
Angle ∠ C = γ = 0.93333333333° = 0°56' = 0.01662897397 rad

Height: ha = 0.41221121874
Height: hb = 0.92202305752
Height: hc = 0.74662126139

Median: ma = 2.97989611054
Median: mb = 43.84660839924
Median: mc = 40.89658015139

Inradius: r = 0.20660450089
Circumradius: R = 957.7700383638

Vertex coordinates: A[31.2; 0] B[0; 0] C[56.48989929957; 0.74662126139]
Centroid: CG[29.23296643319; 0.2498737538]
Coordinates of the circumscribed circle: U[15.6; 957.5733320859]
Coordinates of the inscribed circle: I[31.19769607344; 0.20660450089]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 1.69901594771° = 1°41'25″ = 3.11220938058 rad
∠ B' = β' = 179.2433173856° = 179°14'35″ = 0.01332091081 rad
∠ C' = γ' = 179.0676666667° = 179°4' = 0.01662897397 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 = 0.933° ; ;

2. From angle γ, side b and side c we calculate side 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 0° 56' ; ; ; ; ; ; a**2 -50.593a -333.35 =0 ; ; a=1; b=-50.593; c=-333.35 ; ; D = b**2 - 4ac = 50.593**2 - 4 * 1 * (-333.35) = 3893.08065418 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 50.59 ± sqrt{ 3893.08 } }{ 2 } ; ; a_{1,2} = 25.29664333 ± 31.1972781432 ; ; a_{1} = 56.4939214732 ; ; a_{2} = -5.90063481321 ; ;
 ; ; text{ Factored form: } ; ; (a -56.4939214732) (a +5.90063481321) = 0 ; ; ; ; a > 0 ; ; ; ; a = 56.494 ; ;


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 = 56.49 ; ; b = 25.3 ; ; c = 31.2 ; ;

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

p = a+b+c = 56.49+25.3+31.2 = 112.99 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 112.99 }{ 2 } = 56.5 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 56.5 * (56.5-56.49)(56.5-25.3)(56.5-31.2) } ; ; T = sqrt{ 135.51 } = 11.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11.64 }{ 56.49 } = 0.41 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11.64 }{ 25.3 } = 0.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11.64 }{ 31.2 } = 0.75 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 25.3**2+31.2**2-56.49**2 }{ 2 * 25.3 * 31.2 } ) = 178° 18'35" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 56.49**2+31.2**2-25.3**2 }{ 2 * 56.49 * 31.2 } ) = 0° 45'25" ; ; gamma = 180° - alpha - beta = 180° - 178° 18'35" - 0° 45'25" = 0° 56' ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11.64 }{ 56.5 } = 0.21 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 56.49 }{ 2 * sin 178° 18'35" } = 957.7 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 25.3**2+2 * 31.2**2 - 56.49**2 } }{ 2 } = 2.979 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 31.2**2+2 * 56.49**2 - 25.3**2 } }{ 2 } = 43.846 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 25.3**2+2 * 56.49**2 - 31.2**2 } }{ 2 } = 40.896 ; ;
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