Triangle calculator SSA

Obtuse scalene triangle.

Sides: a = 125 b = 150 c = 253.6788210186

Area: T = 6700.56552644
Perimeter: p = 528.6788210186
Semiperimeter: s = 264.3399105093

Angle ∠ A = α = 20.62108271377° = 20°37'15″ = 0.3659901328 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 134.3799172862° = 134°22'45″ = 2.34553590126 rad

Height: h_a = 107.209904423
Height: h_b = 89.3410870192
Height: h_c = 52.82772827176

Median: m_a = 198.7976547157
Median: m_b = 185.3754801852
Median: m_c = 54.53875230384

Inradius: r = 25.34883693305
Circumradius: R = 177.4655118736

Vertex coordinates: A[253.6788210186; 0] B[0; 0] C[113.288847338; 52.82772827176]
Centroid: CG[122.3222227855; 17.60990942392]
Coordinates of the circumscribed circle: U[126.8399105093; -124.1219739717]
Coordinates of the inscribed circle: I[114.3399105093; 25.34883693305]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.3799172862° = 159°22'45″ = 0.3659901328 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 45.62108271377° = 45°37'15″ = 2.34553590126 rad

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How did we calculate this triangle?

1. Use Law of Cosines

$a = 125 ; ; b = 150 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 150**2 = 125**2 + c**2 -2 * 125 * c * cos (25° ) ; ; ; ; c**2 -226.577c -6875 =0 ; ; p=1; q=-226.577; r=-6875 ; ; D = q**2 - 4pr = 226.577**2 - 4 * 1 * (-6875) = 78837.1128027 ; ; D>0 ; ; ; ;$
$c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 226.58 ± sqrt{ 78837.11 } }{ 2 } ; ; c_{1,2} = 113.28847338 ± 140.389736807 ; ; c_{1} = 253.678210186 ; ; c_{2} = -27.1012634272 ; ; ; ; text{ Factored form: } ; ; (c -253.678210186) (c +27.1012634272) = 0 ; ; ; ; c>0 ; ;$
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.

$a = 125 ; ; b = 150 ; ; c = 253.68 ; ;$

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

$p = a+b+c = 125+150+253.68 = 528.68 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 528.68 }{ 2 } = 264.34 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 264.34 * (264.34-125)(264.34-150)(264.34-253.68) } ; ; T = sqrt{ 44897574.86 } = 6700.57 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6700.57 }{ 125 } = 107.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6700.57 }{ 150 } = 89.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6700.57 }{ 253.68 } = 52.83 ; ;$

6. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 150**2+253.68**2-125**2 }{ 2 * 150 * 253.68 } ) = 20° 37'15" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 125**2+253.68**2-150**2 }{ 2 * 125 * 253.68 } ) = 25° ; ;$
$gamma = 180° - alpha - beta = 180° - 20° 37'15" - 25° = 134° 22'45" ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 6700.57 }{ 264.34 } = 25.35 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 125 }{ 2 * sin 20° 37'15" } = 177.47 ; ;$

9. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 150**2+2 * 253.68**2 - 125**2 } }{ 2 } = 198.797 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 253.68**2+2 * 125**2 - 150**2 } }{ 2 } = 185.375 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 150**2+2 * 125**2 - 253.68**2 } }{ 2 } = 54.538 ; ;$
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