Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 100   b = 120   c = 111.3555287257

Area: T = 5196.152242271
Perimeter: p = 331.3555287257
Semiperimeter: s = 165.6787643628

Angle ∠ A = α = 51.05217244354° = 51°3'6″ = 0.89110206802 rad
Angle ∠ B = β = 68.94882755646° = 68°56'54″ = 1.20333744222 rad
Angle ∠ C = γ = 60° = 1.04771975512 rad

Height: ha = 103.9233048454
Height: hb = 86.60325403784
Height: hc = 93.32656525257

Median: ma = 104.4033065089
Median: mb = 87.17879788708
Median: mc = 95.39439201417

Inradius: r = 31.36330270742
Circumradius: R = 64.29110050733

Vertex coordinates: A[111.3555287257; 0] B[0; 0] C[35.92110604054; 93.32656525257]
Centroid: CG[49.09221158873; 31.10985508419]
Coordinates of the circumscribed circle: U[55.67876436283; 32.14655025366]
Coordinates of the inscribed circle: I[45.67876436283; 31.36330270742]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.9488275565° = 128°56'54″ = 0.89110206802 rad
∠ B' = β' = 111.0521724435° = 111°3'6″ = 1.20333744222 rad
∠ C' = γ' = 120° = 1.04771975512 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 = 100 ; ; b = 120 ; ; gamma = 60° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 100**2+120**2 - 2 * 100 * 120 * cos(60° ) } ; ; c = 111.36 ; ;


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 = 120 ; ; c = 111.36 ; ;

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

p = a+b+c = 100+120+111.36 = 331.36 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 331.36 }{ 2 } = 165.68 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 165.68 * (165.68-100)(165.68-120)(165.68-111.36) } ; ; T = sqrt{ 27000000 } = 5196.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5196.15 }{ 100 } = 103.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5196.15 }{ 120 } = 86.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5196.15 }{ 111.36 } = 93.33 ; ;

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**2-120**2-111.36**2 }{ 2 * 120 * 111.36 } ) = 51° 3'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 120**2-100**2-111.36**2 }{ 2 * 100 * 111.36 } ) = 68° 56'54" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 111.36**2-100**2-120**2 }{ 2 * 120 * 100 } ) = 60° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5196.15 }{ 165.68 } = 31.36 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 51° 3'6" } = 64.29 ; ;




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