Triangle calculator SAS

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


Right scalene triangle.

Sides: a = 178   b = 201   c = 268.4866498729

Area: T = 17889
Perimeter: p = 647.4866498729
Semiperimeter: s = 323.7433249365

Angle ∠ A = α = 41.52772064314° = 41°31'38″ = 0.72547864814 rad
Angle ∠ B = β = 48.47327935686° = 48°28'22″ = 0.84660098454 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 201
Height: hb = 178
Height: hc = 133.2588097406

Median: ma = 219.8232655793
Median: mb = 204.4121961489
Median: mc = 134.2433249365

Inradius: r = 55.25767506353
Circumradius: R = 134.2433249365

Vertex coordinates: A[268.4866498729; 0] B[0; 0] C[118.01096584; 133.2588097406]
Centroid: CG[128.8322052376; 44.41993658021]
Coordinates of the circumscribed circle: U[134.2433249365; -0]
Coordinates of the inscribed circle: I[122.7433249365; 55.25767506353]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.4732793569° = 138°28'22″ = 0.72547864814 rad
∠ B' = β' = 131.5277206431° = 131°31'38″ = 0.84660098454 rad
∠ C' = γ' = 90° = 1.57107963268 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 = 178 ; ; b = 201 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 178**2+201**2 - 2 * 178 * 201 * cos(90° ) } ; ; c = 268.49 ; ;


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 = 178 ; ; b = 201 ; ; c = 268.49 ; ;

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

p = a+b+c = 178+201+268.49 = 647.49 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 647.49 }{ 2 } = 323.74 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 323.74 * (323.74-178)(323.74-201)(323.74-268.49) } ; ; T = sqrt{ 320016321 } = 17889 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17889 }{ 178 } = 201 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17889 }{ 201 } = 178 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17889 }{ 268.49 } = 133.26 ; ;

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{ 178**2-201**2-268.49**2 }{ 2 * 201 * 268.49 } ) = 41° 31'38" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 201**2-178**2-268.49**2 }{ 2 * 178 * 268.49 } ) = 48° 28'22" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 268.49**2-178**2-201**2 }{ 2 * 201 * 178 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17889 }{ 323.74 } = 55.26 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 178 }{ 2 * sin 41° 31'38" } = 134.24 ; ;




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