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 = 20.5   b = 35   c = 52.49900858932

Area: T = 227.3798505803
Perimeter: p = 107.9990085893
Semiperimeter: s = 53.99550429466

Angle ∠ A = α = 14.33216084388° = 14°19'54″ = 0.25501337544 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 140.6688391561° = 140°40'6″ = 2.45551265862 rad

Height: ha = 22.18332688588
Height: hb = 12.99330574744
Height: hc = 8.66436743657

Median: ma = 43.41770710497
Median: mb = 35.79877591273
Median: mc = 11.56881770703

Inradius: r = 4.21110996379
Circumradius: R = 41.40985277052

Vertex coordinates: A[52.49900858932; 0] B[0; 0] C[18.57993096343; 8.66436743657]
Centroid: CG[23.69897985092; 2.88878914552]
Coordinates of the circumscribed circle: U[26.24550429466; -32.02991100008]
Coordinates of the inscribed circle: I[18.99550429466; 4.21110996379]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.6688391561° = 165°40'6″ = 0.25501337544 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 39.33216084388° = 39°19'54″ = 2.45551265862 rad

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How did we calculate this triangle?

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

a = 20.5 ; ; b = 35 ; ; beta = 25° ; ;

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 ; ; ; ; 35**2 = 20.5**2 + c**2 - 2 * 20.5 * c * cos(25° ) ; ; ; ; ; ; c**2 -37.159c -804.75 =0 ; ; a=1; b=-37.159; c=-804.75 ; ; D = b**2 - 4ac = 37.159**2 - 4 * 1 * (-804.75) = 4599.76298594 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 37.16 ± sqrt{ 4599.76 } }{ 2 } ; ; c_{1,2} = 18.57930963 ± 33.910776259 ; ; c_{1} = 52.490085889 ; ; c_{2} = -15.331466629 ; ;
 ; ; text{ Factored form: } ; ; (c -52.490085889) (c +15.331466629) = 0 ; ; ; ; c > 0 ; ; ; ; c = 52.49 ; ;


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 = 20.5 ; ; b = 35 ; ; c = 52.49 ; ;

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

p = a+b+c = 20.5+35+52.49 = 107.99 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 107.99 }{ 2 } = 54 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 54 * (54-20.5)(54-35)(54-52.49) } ; ; T = sqrt{ 51700.98 } = 227.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 227.38 }{ 20.5 } = 22.18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 227.38 }{ 35 } = 12.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 227.38 }{ 52.49 } = 8.66 ; ;

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{ 35**2+52.49**2-20.5**2 }{ 2 * 35 * 52.49 } ) = 14° 19'54" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 20.5**2+52.49**2-35**2 }{ 2 * 20.5 * 52.49 } ) = 25° ; ; gamma = 180° - alpha - beta = 180° - 14° 19'54" - 25° = 140° 40'6" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 227.38 }{ 54 } = 4.21 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20.5 }{ 2 * sin 14° 19'54" } = 41.41 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 35**2+2 * 52.49**2 - 20.5**2 } }{ 2 } = 43.417 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 52.49**2+2 * 20.5**2 - 35**2 } }{ 2 } = 35.798 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 35**2+2 * 20.5**2 - 52.49**2 } }{ 2 } = 11.568 ; ;
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