Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 49   b = 31   c = 60.65993560088

Area: T = 755.3399379532
Perimeter: p = 140.6599356009
Semiperimeter: s = 70.33296780044

Angle ∠ A = α = 53.45325997546° = 53°27'9″ = 0.93329238595 rad
Angle ∠ B = β = 30.54774002454° = 30°32'51″ = 0.53331527122 rad
Angle ∠ C = γ = 96° = 1.67655160819 rad

Height: ha = 30.83301787564
Height: hb = 48.7321572873
Height: hc = 24.90442993276

Median: ma = 41.47332291449
Median: mb = 52.91552977475
Median: mc = 27.58882335815

Inradius: r = 10.74399806307
Circumradius: R = 30.49767423499

Vertex coordinates: A[60.65993560088; 0] B[0; 0] C[42.19992402183; 24.90442993276]
Centroid: CG[34.28661987424; 8.30114331092]
Coordinates of the circumscribed circle: U[30.33296780044; -3.18877776125]
Coordinates of the inscribed circle: I[39.33296780044; 10.74399806307]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.5477400245° = 126°32'51″ = 0.93329238595 rad
∠ B' = β' = 149.4532599755° = 149°27'9″ = 0.53331527122 rad
∠ C' = γ' = 84° = 1.67655160819 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 = 49 ; ; b = 31 ; ; gamma = 96° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 49**2+31**2 - 2 * 49 * 31 * cos(96° ) } ; ; c = 60.66 ; ;


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 = 49 ; ; b = 31 ; ; c = 60.66 ; ;

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

p = a+b+c = 49+31+60.66 = 140.66 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 140.66 }{ 2 } = 70.33 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 70.33 * (70.33-49)(70.33-31)(70.33-60.66) } ; ; T = sqrt{ 570537.58 } = 755.34 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 755.34 }{ 49 } = 30.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 755.34 }{ 31 } = 48.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 755.34 }{ 60.66 } = 24.9 ; ;

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{ 49**2-31**2-60.66**2 }{ 2 * 31 * 60.66 } ) = 53° 27'9" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 31**2-49**2-60.66**2 }{ 2 * 49 * 60.66 } ) = 30° 32'51" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 60.66**2-49**2-31**2 }{ 2 * 31 * 49 } ) = 96° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 755.34 }{ 70.33 } = 10.74 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 49 }{ 2 * sin 53° 27'9" } = 30.5 ; ;




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