Triangle calculator SSA

Please enter two sides and a non-included angle
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Obtuse scalene triangle.

Sides: a = 100   b = 125   c = 48.2288427765

Area: T = 2265.995488415
Perimeter: p = 273.2288427765
Semiperimeter: s = 136.6144213882

Angle ∠ A = α = 48.74325524574° = 48°44'33″ = 0.85107180262 rad
Angle ∠ B = β = 110° = 1.92198621772 rad
Angle ∠ C = γ = 21.25774475426° = 21°15'27″ = 0.37110124502 rad

Height: ha = 45.32198976829
Height: hb = 36.25659181463
Height: hc = 93.96992620786

Median: ma = 80.47704332183
Median: mb = 47.5055164165
Median: mc = 110.5943872745

Inradius: r = 16.58768164062
Circumradius: R = 66.51111107797

Vertex coordinates: A[48.2288427765; 0] B[0; 0] C[-34.20220143326; 93.96992620786]
Centroid: CG[4.67554711442; 31.32330873595]
Coordinates of the circumscribed circle: U[24.11442138825; 61.98657446998]
Coordinates of the inscribed circle: I[11.61442138825; 16.58768164062]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.2577447543° = 131°15'27″ = 0.85107180262 rad
∠ B' = β' = 70° = 1.92198621772 rad
∠ C' = γ' = 158.7432552457° = 158°44'33″ = 0.37110124502 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 125 ; ; beta = 110° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 125**2 = 100**2 + c**2 -2 * 125 * c * cos (110° ) ; ; ; ; c**2 +68.404c -5625 =0 ; ; p=1; q=68.4040286651; r=-5625 ; ; D = q**2 - 4pr = 68.404**2 - 4 * 1 * (-5625) = 27179.1111376 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ -68.4 ± sqrt{ 27179.11 } }{ 2 } ; ; c_{1,2} = -34.2020143326 ± 82.4304420976 ; ;
c_{1} = 48.228427765 ; ; c_{2} = -116.63245643 ; ; ; ; (c -48.228427765) (c +116.63245643) = 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 = 100 ; ; b = 125 ; ; c = 48.23 ; ;

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

p = a+b+c = 100+125+48.23 = 273.23 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 273.23 }{ 2 } = 136.61 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 136.61 * (136.61-100)(136.61-125)(136.61-48.23) } ; ; T = sqrt{ 5134732.81 } = 2265.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2265.99 }{ 100 } = 45.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2265.99 }{ 125 } = 36.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2265.99 }{ 48.23 } = 93.97 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 100**2-125**2-48.23**2 }{ 2 * 125 * 48.23 } ) = 48° 44'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 125**2-100**2-48.23**2 }{ 2 * 100 * 48.23 } ) = 110° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 48.23**2-100**2-125**2 }{ 2 * 125 * 100 } ) = 21° 15'27" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2265.99 }{ 136.61 } = 16.59 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 48° 44'33" } = 66.51 ; ;




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