Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 100   b = 120   c = 86.19441833973

Area: T = 4242.641068712
Perimeter: p = 306.1944183397
Semiperimeter: s = 153.0977091699

Angle ∠ A = α = 55.12113330792° = 55°7'17″ = 0.96220487503 rad
Angle ∠ B = β = 79.87986669208° = 79°52'43″ = 1.39441457399 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 84.85328137424
Height: hb = 70.71106781187
Height: hc = 98.44437817008

Median: ma = 91.73217754421
Median: mb = 71.51772610337
Median: mc = 101.6998774266

Inradius: r = 27.71220919806
Circumradius: R = 60.9488491579

Vertex coordinates: A[86.19441833973; 0] B[0; 0] C[17.5733327643; 98.44437817008]
Centroid: CG[34.58991703468; 32.81545939003]
Coordinates of the circumscribed circle: U[43.09770916986; 43.09770916986]
Coordinates of the inscribed circle: I[33.09770916986; 27.71220919806]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.8798666921° = 124°52'43″ = 0.96220487503 rad
∠ B' = β' = 100.1211333079° = 100°7'17″ = 1.39441457399 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 = 100 ; ; b = 120 ; ; gamma = 45° ; ; ; ; 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(45° ) } ; ; c = 86.19 ; ;


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 = 86.19 ; ;

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

p = a+b+c = 100+120+86.19 = 306.19 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 306.19 }{ 2 } = 153.1 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 153.1 * (153.1-100)(153.1-120)(153.1-86.19) } ; ; T = sqrt{ 18000000 } = 4242.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4242.64 }{ 100 } = 84.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4242.64 }{ 120 } = 70.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4242.64 }{ 86.19 } = 98.44 ; ;

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**2-86.19**2 }{ 2 * 120 * 86.19 } ) = 55° 7'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 120**2-100**2-86.19**2 }{ 2 * 100 * 86.19 } ) = 79° 52'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 86.19**2-100**2-120**2 }{ 2 * 120 * 100 } ) = 45° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4242.64 }{ 153.1 } = 27.71 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 55° 7'17" } = 60.95 ; ;




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