Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 218.5   b = 213.3   c = 212.5465504729

Area: T = 19966.20772321
Perimeter: p = 644.3465504729
Semiperimeter: s = 322.1732752365

Angle ∠ A = α = 61.74403640349° = 61°44'25″ = 1.07875726338 rad
Angle ∠ B = β = 59.32996359651° = 59°17'59″ = 1.03549738928 rad
Angle ∠ C = γ = 58.96° = 58°57'36″ = 1.0299046127 rad

Height: ha = 182.7577045602
Height: hb = 187.2122444745
Height: hc = 187.8777012572

Median: ma = 182.7598524535
Median: mb = 187.3099098258
Median: mc = 187.9511249277

Inradius: r = 61.97436060406
Circumradius: R = 124.0333401857

Vertex coordinates: A[212.5465504729; 0] B[0; 0] C[111.5554821263; 187.8777012572]
Centroid: CG[108.0333441997; 62.62656708574]
Coordinates of the circumscribed circle: U[106.2732752365; 63.95661324739]
Coordinates of the inscribed circle: I[108.8732752365; 61.97436060406]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 118.2659635965° = 118°15'35″ = 1.07875726338 rad
∠ B' = β' = 120.7700364035° = 120°42'1″ = 1.03549738928 rad
∠ C' = γ' = 121.04° = 121°2'24″ = 1.0299046127 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 = 218.5 ; ; b = 213.3 ; ; gamma = 58° 57'36" ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 218.5**2+213.3**2 - 2 * 218.5 * 213.3 * cos(58° 57'36") } ; ; c = 212.55 ; ;


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 = 218.5 ; ; b = 213.3 ; ; c = 212.55 ; ;

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

p = a+b+c = 218.5+213.3+212.55 = 644.35 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 644.35 }{ 2 } = 322.17 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 322.17 * (322.17-218.5)(322.17-213.3)(322.17-212.55) } ; ; T = sqrt{ 398649431.23 } = 19966.21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 19966.21 }{ 218.5 } = 182.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 19966.21 }{ 213.3 } = 187.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 19966.21 }{ 212.55 } = 187.88 ; ;

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{ 218.5**2-213.3**2-212.55**2 }{ 2 * 213.3 * 212.55 } ) = 61° 44'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 213.3**2-218.5**2-212.55**2 }{ 2 * 218.5 * 212.55 } ) = 59° 17'59" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 212.55**2-218.5**2-213.3**2 }{ 2 * 213.3 * 218.5 } ) = 58° 57'36" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 19966.21 }{ 322.17 } = 61.97 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 218.5 }{ 2 * sin 61° 44'25" } = 124.03 ; ;




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