Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 104.55   b = 123.84   c = 177.9077025411

Area: T = 6332.269933618
Perimeter: p = 406.2977025411
Semiperimeter: s = 203.1498512706

Angle ∠ A = α = 35.08773485468° = 35°5'14″ = 0.61223897579 rad
Angle ∠ B = β = 42.91326514532° = 42°54'46″ = 0.74989670586 rad
Angle ∠ C = γ = 102° = 1.7880235837 rad

Height: ha = 121.1343798875
Height: hb = 102.2655331657
Height: hc = 71.18662763322

Median: ma = 144.0876612912
Median: mb = 132.124388011
Median: mc = 72.25550802874

Inradius: r = 31.17106408866
Circumradius: R = 90.94107870948

Vertex coordinates: A[177.9077025411; 0] B[0; 0] C[76.5721643328; 71.18662763322]
Centroid: CG[84.82662229131; 23.72987587774]
Coordinates of the circumscribed circle: U[88.95435127056; -18.90876528092]
Coordinates of the inscribed circle: I[79.30985127056; 31.17106408866]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.9132651453° = 144°54'46″ = 0.61223897579 rad
∠ B' = β' = 137.0877348547° = 137°5'14″ = 0.74989670586 rad
∠ C' = γ' = 78° = 1.7880235837 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 = 104.55 ; ; b = 123.84 ; ; gamma = 102° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 104.55**2+123.84**2 - 2 * 104.55 * 123.84 * cos(102° ) } ; ; c = 177.91 ; ;


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 = 104.55 ; ; b = 123.84 ; ; c = 177.91 ; ;

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

p = a+b+c = 104.55+123.84+177.91 = 406.3 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 406.3 }{ 2 } = 203.15 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 203.15 * (203.15-104.55)(203.15-123.84)(203.15-177.91) } ; ; T = sqrt{ 40097634.95 } = 6332.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6332.27 }{ 104.55 } = 121.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6332.27 }{ 123.84 } = 102.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6332.27 }{ 177.91 } = 71.19 ; ;

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{ 104.55**2-123.84**2-177.91**2 }{ 2 * 123.84 * 177.91 } ) = 35° 5'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 123.84**2-104.55**2-177.91**2 }{ 2 * 104.55 * 177.91 } ) = 42° 54'46" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 177.91**2-104.55**2-123.84**2 }{ 2 * 123.84 * 104.55 } ) = 102° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6332.27 }{ 203.15 } = 31.17 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 104.55 }{ 2 * sin 35° 5'14" } = 90.94 ; ;




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