Triangle calculator SSA

Please enter two sides and a non-included angle
°


Acute scalene triangle.

Sides: a = 11.4   b = 12.8   c = 14.4330039195

Area: T = 69.75329934209
Perimeter: p = 38.6330039195
Semiperimeter: s = 19.31550195975

Angle ∠ A = α = 49.05109636827° = 49°3'3″ = 0.85661008175 rad
Angle ∠ B = β = 58° = 1.01222909662 rad
Angle ∠ C = γ = 72.94990363173° = 72°56'57″ = 1.27332008699 rad

Height: ha = 12.23773672668
Height: hb = 10.8998905222
Height: hc = 9.66877482962

Median: ma = 12.39112475395
Median: mb = 11.32195854865
Median: mc = 9.7398762355

Inradius: r = 3.61113343333
Circumradius: R = 7.54767417815

Vertex coordinates: A[14.4330039195; 0] B[0; 0] C[6.04110796123; 9.66877482962]
Centroid: CG[6.82437062691; 3.22325827654]
Coordinates of the circumscribed circle: U[7.21550195975; 2.21328722792]
Coordinates of the inscribed circle: I[6.51550195975; 3.61113343333]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.9499036317° = 130°56'57″ = 0.85661008175 rad
∠ B' = β' = 122° = 1.01222909662 rad
∠ C' = γ' = 107.0510963683° = 107°3'3″ = 1.27332008699 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 11.4 ; ; b = 12.8 ; ; beta = 58° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 12.8**2 = 11.4**2 + c**2 -2 * 12.8 * c * cos (58° ) ; ; ; ; c**2 -12.082c -33.88 =0 ; ; p=1; q=-12.0821592245; r=-33.88 ; ; D = q**2 - 4pr = 12.082**2 - 4 * 1 * (-33.88) = 281.498571527 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 12.08 ± sqrt{ 281.5 } }{ 2 } ; ; c_{1,2} = 6.04107961226 ± 8.38895958279 ; ;
c_{1} = 14.430039195 ; ; c_{2} = -2.34787997053 ; ; ; ; (c -14.430039195) (c +2.34787997053) = 0 ; ; ; ; c>0 ; ;


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 = 11.4 ; ; b = 12.8 ; ; c = 14.43 ; ;

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

p = a+b+c = 11.4+12.8+14.43 = 38.63 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38.63 }{ 2 } = 19.32 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.32 * (19.32-11.4)(19.32-12.8)(19.32-14.43) } ; ; T = sqrt{ 4865.48 } = 69.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 69.75 }{ 11.4 } = 12.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 69.75 }{ 12.8 } = 10.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 69.75 }{ 14.43 } = 9.67 ; ;

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{ 11.4**2-12.8**2-14.43**2 }{ 2 * 12.8 * 14.43 } ) = 49° 3'3" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.8**2-11.4**2-14.43**2 }{ 2 * 11.4 * 14.43 } ) = 58° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14.43**2-11.4**2-12.8**2 }{ 2 * 12.8 * 11.4 } ) = 72° 56'57" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 69.75 }{ 19.32 } = 3.61 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.4 }{ 2 * sin 49° 3'3" } = 7.55 ; ;




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