Triangle calculator

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

You have entered side a, c and angle α.

Obtuse scalene triangle.

Sides: a = 46   b = 63.99881197287   c = 37

Area: T = 837.1989832202
Perimeter: p = 146.9988119729
Semiperimeter: s = 73.49990598643

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 100.3366211802° = 100°20'10″ = 1.75111972549 rad
Angle ∠ C = γ = 34.66437881979° = 34°39'50″ = 0.60549972353 rad

Height: ha = 36.43995579218
Height: hb = 26.16329509039
Height: hc = 45.25435044433

Median: ma = 46.9440171116
Median: mb = 26.80659726143
Median: mc = 52.57702355369

Inradius: r = 11.39904835483
Circumradius: R = 32.52769119346

Vertex coordinates: A[37; 0] B[0; 0] C[-8.25435044433; 45.25435044433]
Centroid: CG[9.58221651856; 15.08545014811]
Coordinates of the circumscribed circle: U[18.5; 26.75435044433]
Coordinates of the inscribed circle: I[9.50109401357; 11.39904835483]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 79.66437881979° = 79°39'50″ = 1.75111972549 rad
∠ C' = γ' = 145.3366211802° = 145°20'10″ = 0.60549972353 rad

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How did we calculate this triangle?

1. Input data entered: side a, c and angle α.

a = 46 ; ; c = 37 ; ; alpha = 45° ; ;

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 46**2 = 37**2 + b**2 - 2 * 37 * b * cos 45° ; ; ; ; ; ; b**2 -52.326b -747 =0 ; ; a=1; b=-52.326; c=-747 ; ; D = b**2 - 4ac = 52.326**2 - 4 * 1 * (-747) = 5726 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 52.33 ± sqrt{ 5726 } }{ 2 } ; ; b_{1,2} = 26.1629509 ± 37.8351688248 ; ; b_{1} = 63.9981197248 ; ; b_{2} = -11.6722179248 ; ; ; ; text{ Factored form: } ; ;
(b -63.9981197248) (b +11.6722179248) = 0 ; ; ; ; b > 0 ; ; ; ; b = 63.998 ; ;


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 = 46 ; ; b = 64 ; ; c = 37 ; ;

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

p = a+b+c = 46+64+37 = 147 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 147 }{ 2 } = 73.5 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 73.5 * (73.5-46)(73.5-64)(73.5-37) } ; ; T = sqrt{ 700886.82 } = 837.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 837.19 }{ 46 } = 36.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 837.19 }{ 64 } = 26.16 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 837.19 }{ 37 } = 45.25 ; ;

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{ 64**2+37**2-46**2 }{ 2 * 64 * 37 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 46**2+37**2-64**2 }{ 2 * 46 * 37 } ) = 100° 20'10" ; ; gamma = 180° - alpha - beta = 180° - 45° - 100° 20'10" = 34° 39'50" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 837.19 }{ 73.5 } = 11.39 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 46 }{ 2 * sin 45° } = 32.53 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 64**2+2 * 37**2 - 46**2 } }{ 2 } = 46.94 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 37**2+2 * 46**2 - 64**2 } }{ 2 } = 26.806 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 64**2+2 * 46**2 - 37**2 } }{ 2 } = 52.57 ; ;
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