Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side b, c and angle α.

Obtuse scalene triangle.

Sides: a = 92.99004311645   b = 186.01   c = 156.49

Area: T = 7259.57700202909
Perimeter: p = 435.44004311645
Semiperimeter: s = 217.77002155823

Angle ∠ A = α = 29.92° = 29°55'12″ = 0.52222025122 rad
Angle ∠ B = β = 92.91878974245° = 92°55'4″ = 1.62217232441 rad
Angle ∠ C = γ = 57.16221025755° = 57°9'44″ = 0.99876668973 rad

Height: ha = 156.28771114653
Height: hb = 78.05656961485
Height: hc = 92.78799862009

Median: ma = 165.49895693763
Median: mb = 88.93774784907
Median: mc = 124.47701774735

Inradius: r = 33.34766367999
Circumradius: R = 93.12657370716

Vertex coordinates: A[156.49; 0] B[0; 0] C[-4.72990877674; 92.78799862009]
Centroid: CG[50.58769707442; 30.9276662067]
Coordinates of the circumscribed circle: U[78.245; 50.49987413718]
Coordinates of the inscribed circle: I[31.69902155823; 33.34766367999]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.08° = 150°4'48″ = 0.52222025122 rad
∠ B' = β' = 87.08221025755° = 87°4'56″ = 1.62217232441 rad
∠ C' = γ' = 122.83878974245° = 122°50'16″ = 0.99876668973 rad

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How did we calculate this triangle?

1. Input data entered: side b, c and angle α.

b = 186.01 ; ; c = 156.49 ; ; alpha = 29.92° ; ;

2. Calculation of the third side a of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 186.01**2+156.49**2 - 2 * 186.01 * 156.49 * cos(29° 55'12") } ; ; a = 92.9 ; ;


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 = 92.9 ; ; b = 186.01 ; ; c = 156.49 ; ;

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

p = a+b+c = 92.9+186.01+156.49 = 435.4 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 435.4 }{ 2 } = 217.7 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 217.7 * (217.7-92.9)(217.7-186.01)(217.7-156.49) } ; ; T = sqrt{ 52701356.88 } = 7259.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7259.57 }{ 92.9 } = 156.29 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7259.57 }{ 186.01 } = 78.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7259.57 }{ 156.49 } = 92.78 ; ;

7. 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 186.01**2+156.49**2-92.9**2 }{ 2 * 186.01 * 156.49 } ) = 29° 55'12" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 92.9**2+156.49**2-186.01**2 }{ 2 * 92.9 * 156.49 } ) = 92° 55'4" ; ; gamma = 180° - alpha - beta = 180° - 29° 55'12" - 92° 55'4" = 57° 9'44" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7259.57 }{ 217.7 } = 33.35 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 92.9 }{ 2 * sin 29° 55'12" } = 93.13 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 186.01**2+2 * 156.49**2 - 92.9**2 } }{ 2 } = 165.49 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 156.49**2+2 * 92.9**2 - 186.01**2 } }{ 2 } = 88.937 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 186.01**2+2 * 92.9**2 - 156.49**2 } }{ 2 } = 124.47 ; ;
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