Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 220   b = 510   c = 614.6989688817

Area: T = 53354.27105642
Perimeter: p = 1344.698968882
Semiperimeter: s = 672.3454844408

Angle ∠ A = α = 19.99004593484° = 19°54'2″ = 0.34773285383 rad
Angle ∠ B = β = 52.10995406516° = 52°5'58″ = 0.90993085231 rad
Angle ∠ C = γ = 108° = 1.88549555922 rad

Height: ha = 485.039882331
Height: hb = 209.2322433585
Height: hc = 173.5977415199

Median: ma = 553.9660022717
Median: mb = 384.8333349346
Median: mc = 244.5188192811

Inradius: r = 79.35655137782
Circumradius: R = 323.161149371

Vertex coordinates: A[614.6989688817; 0] B[0; 0] C[135.1444135781; 173.5977415199]
Centroid: CG[249.9454608199; 57.86658050665]
Coordinates of the circumscribed circle: U[307.3454844408; -99.86223934839]
Coordinates of the inscribed circle: I[162.3454844408; 79.35655137782]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.1099540652° = 160°5'58″ = 0.34773285383 rad
∠ B' = β' = 127.9900459348° = 127°54'2″ = 0.90993085231 rad
∠ C' = γ' = 72° = 1.88549555922 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 = 220 ; ; b = 510 ; ; gamma = 108° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 220**2+510**2 - 2 * 220 * 510 * cos(108° ) } ; ; c = 614.69 ; ;


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 = 220 ; ; b = 510 ; ; c = 614.69 ; ;

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

p = a+b+c = 220+510+614.69 = 1344.69 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1344.69 }{ 2 } = 672.34 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 672.34 * (672.34-220)(672.34-510)(672.34-614.69) } ; ; T = sqrt{ 2846678187.43 } = 53354.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 53354.27 }{ 220 } = 485.04 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 53354.27 }{ 510 } = 209.23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 53354.27 }{ 614.69 } = 173.6 ; ;

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{ 220**2-510**2-614.69**2 }{ 2 * 510 * 614.69 } ) = 19° 54'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 510**2-220**2-614.69**2 }{ 2 * 220 * 614.69 } ) = 52° 5'58" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 614.69**2-220**2-510**2 }{ 2 * 510 * 220 } ) = 108° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 53354.27 }{ 672.34 } = 79.36 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 220 }{ 2 * sin 19° 54'2" } = 323.16 ; ;




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