Triangle calculator

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

You have entered side b, angle α, angle β and angle γ.

Obtuse scalene triangle.

Sides: a = 798.9454559674   b = 582   c = 436.998773684

Area: T = 124233.1109626
Perimeter: p = 1817.942229651
Semiperimeter: s = 908.9711148257

Angle ∠ A = α = 102.33° = 102°19'48″ = 1.78659954236 rad
Angle ∠ B = β = 45.37° = 45°22'12″ = 0.79218558816 rad
Angle ∠ C = γ = 32.3° = 32°18' = 0.56437413484 rad

Height: ha = 310.9933067345
Height: hb = 426.9187902494
Height: hc = 568.5765528671

Median: ma = 324.4499393039
Median: mb = 574.4220330175
Median: mc = 663.9109970716

Inradius: r = 136.6744425656
Circumradius: R = 408.9044103574

Vertex coordinates: A[436.998773684; 0] B[0; 0] C[561.2799144125; 568.5765528671]
Centroid: CG[332.7598960322; 189.5255176224]
Coordinates of the circumscribed circle: U[218.499886842; 345.6311032199]
Coordinates of the inscribed circle: I[326.9711148257; 136.6744425656]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 77.67° = 77°40'12″ = 1.78659954236 rad
∠ B' = β' = 134.63° = 134°37'48″ = 0.79218558816 rad
∠ C' = γ' = 147.7° = 147°42' = 0.56437413484 rad

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How did we calculate this triangle?

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

b = 582 ; ; alpha = 102.33° ; ; beta = 45.37° ; ; gamma = 32.3° ; ;

2. From angle α, angle β and side b we calculate side a - By using the law of sines, we calculate unknown side a:

 fraction{ a }{ b } = fraction{ sin alpha }{ sin beta } ; ; ; ; a = b * fraction{ sin alpha }{ sin beta } ; ; ; ; a = 582 * fraction{ sin 102° 19'48" }{ sin 45° 22'12" } = 798.94 ; ;

3. Calculation of the third side c of the triangle using a Law of Cosines

c**2 = b**2+a**2 - 2ba cos gamma ; ; c = sqrt{ b**2+a**2 - 2ba cos gamma } ; ; c = sqrt{ 582**2+798.94**2 - 2 * 582 * 798.94 * cos 32° 18' } ; ; c = 437 ; ;


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 = 798.94 ; ; b = 582 ; ; c = 437 ; ;

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

p = a+b+c = 798.94+582+437 = 1817.94 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1817.94 }{ 2 } = 908.97 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 908.97 * (908.97-798.94)(908.97-582)(908.97-437) } ; ; T = sqrt{ 15433865527.3 } = 124233.11 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 124233.11 }{ 798.94 } = 310.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 124233.11 }{ 582 } = 426.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 124233.11 }{ 437 } = 568.58 ; ;

8. 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{ 582**2+437**2-798.94**2 }{ 2 * 582 * 437 } ) = 102° 19'48" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 798.94**2+437**2-582**2 }{ 2 * 798.94 * 437 } ) = 45° 22'12" ; ; gamma = 180° - alpha - beta = 180° - 102° 19'48" - 45° 22'12" = 32° 18' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 124233.11 }{ 908.97 } = 136.67 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 798.94 }{ 2 * sin 102° 19'48" } = 408.9 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 582**2+2 * 437**2 - 798.94**2 } }{ 2 } = 324.449 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 437**2+2 * 798.94**2 - 582**2 } }{ 2 } = 574.42 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 582**2+2 * 798.94**2 - 437**2 } }{ 2 } = 663.91 ; ;
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