Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 180   b = 329   c = 350.9055191392

Area: T = 29284.76597141
Perimeter: p = 859.9055191392
Semiperimeter: s = 429.9532595696

Angle ∠ A = α = 30.48657901396° = 30°29'9″ = 0.5322077413 rad
Angle ∠ B = β = 68.01442098604° = 68°51″ = 1.18770719002 rad
Angle ∠ C = γ = 81.5° = 81°30' = 1.42224433404 rad

Height: ha = 325.3866219046
Height: hb = 178.0232855405
Height: hc = 166.9109811724

Median: ma = 328.0065680855
Median: mb = 225.1822096697
Median: mc = 198.8398845962

Inradius: r = 68.11216011563
Circumradius: R = 177.4011194658

Vertex coordinates: A[350.9055191392; 0] B[0; 0] C[67.38877937773; 166.9109811724]
Centroid: CG[139.4310995056; 55.63766039081]
Coordinates of the circumscribed circle: U[175.4532595696; 26.22215661161]
Coordinates of the inscribed circle: I[100.9532595696; 68.11216011563]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.514420986° = 149°30'51″ = 0.5322077413 rad
∠ B' = β' = 111.986579014° = 111°59'9″ = 1.18770719002 rad
∠ C' = γ' = 98.5° = 98°30' = 1.42224433404 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 = 180 ; ; b = 329 ; ; gamma = 81° 30' ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 180**2+329**2 - 2 * 180 * 329 * cos(81° 30') } ; ; c = 350.91 ; ;


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 = 180 ; ; b = 329 ; ; c = 350.91 ; ;

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

p = a+b+c = 180+329+350.91 = 859.91 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 859.91 }{ 2 } = 429.95 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 429.95 * (429.95-180)(429.95-329)(429.95-350.91) } ; ; T = sqrt{ 857597151.52 } = 29284.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 29284.76 }{ 180 } = 325.39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 29284.76 }{ 329 } = 178.02 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 29284.76 }{ 350.91 } = 166.91 ; ;

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{ 180**2-329**2-350.91**2 }{ 2 * 329 * 350.91 } ) = 30° 29'9" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 329**2-180**2-350.91**2 }{ 2 * 180 * 350.91 } ) = 68° 51" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 350.91**2-180**2-329**2 }{ 2 * 329 * 180 } ) = 81° 30' ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 29284.76 }{ 429.95 } = 68.11 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 180 }{ 2 * sin 30° 29'9" } = 177.4 ; ;

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