Triangle calculator SAS

Please enter two sides of the triangle and the included angle
°


Obtuse scalene triangle.

Sides: a = 41.5   b = 65   c = 46.17880133343

Area: T = 953.7110271125
Perimeter: p = 152.6788013334
Semiperimeter: s = 76.33990066671

Angle ∠ A = α = 39.4555156382° = 39°27'19″ = 0.68986223858 rad
Angle ∠ B = β = 95.5454843618° = 95°32'41″ = 1.66875721044 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 45.96219407771
Height: hb = 29.34549314192
Height: hc = 41.30658164379

Median: ma = 52.42327236773
Median: mb = 29.51440552576
Median: mc = 49.40216474536

Inradius: r = 12.49330924931
Circumradius: R = 32.65327863704

Vertex coordinates: A[46.17880133343; 0] B[0; 0] C[-4.01099287276; 41.30658164379]
Centroid: CG[14.05660282022; 13.76986054793]
Coordinates of the circumscribed circle: U[23.08990066671; 23.08990066671]
Coordinates of the inscribed circle: I[11.33990066671; 12.49330924931]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.5454843618° = 140°32'41″ = 0.68986223858 rad
∠ B' = β' = 84.4555156382° = 84°27'19″ = 1.66875721044 rad
∠ C' = γ' = 135° = 0.78553981634 rad

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How did we calculate this triangle?

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

a = 41.5 ; ; b = 65 ; ; gamma = 45° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 41.5**2+65**2 - 2 * 41.5 * 65 * cos(45° ) } ; ; c = 46.18 ; ;


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 = 41.5 ; ; b = 65 ; ; c = 46.18 ; ;

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

p = a+b+c = 41.5+65+46.18 = 152.68 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 152.68 }{ 2 } = 76.34 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 76.34 * (76.34-41.5)(76.34-65)(76.34-46.18) } ; ; T = sqrt{ 909563.28 } = 953.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 953.71 }{ 41.5 } = 45.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 953.71 }{ 65 } = 29.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 953.71 }{ 46.18 } = 41.31 ; ;

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{ 41.5**2-65**2-46.18**2 }{ 2 * 65 * 46.18 } ) = 39° 27'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 65**2-41.5**2-46.18**2 }{ 2 * 41.5 * 46.18 } ) = 95° 32'41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 46.18**2-41.5**2-65**2 }{ 2 * 65 * 41.5 } ) = 45° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 953.71 }{ 76.34 } = 12.49 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 41.5 }{ 2 * sin 39° 27'19" } = 32.65 ; ;




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