Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 146.2   b = 209.3   c = 191.412155857

Area: T = 13533.9398542
Perimeter: p = 546.912155857
Semiperimeter: s = 273.4565779285

Angle ∠ A = α = 42.50440454849° = 42°30'15″ = 0.74218355391 rad
Angle ∠ B = β = 75.29659545151° = 75°17'45″ = 1.31441623197 rad
Angle ∠ C = γ = 62.2° = 62°12' = 1.08655947947 rad

Height: ha = 185.1432798113
Height: hb = 129.3265738576
Height: hc = 141.4121925624

Median: ma = 186.75987411
Median: mb = 134.368811332
Median: mc = 153.0711450021

Inradius: r = 49.4922238114
Circumradius: R = 108.193335026

Vertex coordinates: A[191.412155857; 0] B[0; 0] C[37.1099396267; 141.4121925624]
Centroid: CG[76.17436516123; 47.13773085413]
Coordinates of the circumscribed circle: U[95.70657792849; 50.46599331349]
Coordinates of the inscribed circle: I[64.15657792849; 49.4922238114]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.4965954515° = 137°29'45″ = 0.74218355391 rad
∠ B' = β' = 104.7044045485° = 104°42'15″ = 1.31441623197 rad
∠ C' = γ' = 117.8° = 117°48' = 1.08655947947 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 = 146.2 ; ; b = 209.3 ; ; gamma = 62° 12' ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 146.2**2+209.3**2 - 2 * 146.2 * 209.3 * cos(62° 12') } ; ; c = 191.41 ; ;


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 = 146.2 ; ; b = 209.3 ; ; c = 191.41 ; ;

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

p = a+b+c = 146.2+209.3+191.41 = 546.91 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 546.91 }{ 2 } = 273.46 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 273.46 * (273.46-146.2)(273.46-209.3)(273.46-191.41) } ; ; T = sqrt{ 183167492.46 } = 13533.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13533.94 }{ 146.2 } = 185.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13533.94 }{ 209.3 } = 129.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13533.94 }{ 191.41 } = 141.41 ; ;

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{ 146.2**2-209.3**2-191.41**2 }{ 2 * 209.3 * 191.41 } ) = 42° 30'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 209.3**2-146.2**2-191.41**2 }{ 2 * 146.2 * 191.41 } ) = 75° 17'45" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 191.41**2-146.2**2-209.3**2 }{ 2 * 209.3 * 146.2 } ) = 62° 12' ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13533.94 }{ 273.46 } = 49.49 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 146.2 }{ 2 * sin 42° 30'15" } = 108.19 ; ;




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