Triangle calculator

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

You have entered side a, b and angle α.

Obtuse scalene triangle.

Sides: a = 100   b = 90   c = 168.6954510132

Area: T = 3680.312246275
Perimeter: p = 358.6954510132
Semiperimeter: s = 179.3477255066

Angle ∠ A = α = 29° = 0.50661454831 rad
Angle ∠ B = β = 25.8769868638° = 25°52'12″ = 0.45215143848 rad
Angle ∠ C = γ = 125.1330131362° = 125°7'48″ = 2.18439327857 rad

Height: ha = 73.6066249255
Height: hb = 81.78547213945
Height: hc = 43.63328658222

Median: ma = 125.6144166694
Median: mb = 131.1643710203
Median: mc = 43.99547788138

Inradius: r = 20.52105954304
Circumradius: R = 103.1333266981

Vertex coordinates: A[168.6954510132; 0] B[0; 0] C[89.97987364889; 43.63328658222]
Centroid: CG[86.22444155401; 14.54442886074]
Coordinates of the circumscribed circle: U[84.34772550657; -59.34765358814]
Coordinates of the inscribed circle: I[89.34772550657; 20.52105954304]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151° = 0.50661454831 rad
∠ B' = β' = 154.1330131362° = 154°7'48″ = 0.45215143848 rad
∠ C' = γ' = 54.8769868638° = 54°52'12″ = 2.18439327857 rad

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How did we calculate this triangle?

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

a = 100 ; ; b = 90 ; ; alpha = 29° ; ;

2. From angle α, side b and side a we calculate side c - by using the law of cosines and quadratic equation:

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 100**2 = 90**2 + c**2 - 2 * 90 * c * cos 29° ; ; ; ; ; ; c**2 -157.432c -1900 =0 ; ; p=1; q=-157.432; r=-1900 ; ; D = q**2 - 4pr = 157.432**2 - 4 * 1 * (-1900) = 32384.6920806 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 157.43 ± sqrt{ 32384.69 } }{ 2 } ; ; c_{1,2} = 78.71577364 ± 89.9787364889 ; ; c_{1} = 168.694510129 ; ; c_{2} = -11.2629628489 ; ;
 ; ; text{ Factored form: } ; ; (c -168.694510129) (c +11.2629628489) = 0 ; ; ; ; c > 0 ; ; ; ; c = 168.695 ; ;
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 = 100 ; ; b = 90 ; ; c = 168.69 ; ;

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

p = a+b+c = 100+90+168.69 = 358.69 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 358.69 }{ 2 } = 179.35 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 179.35 * (179.35-100)(179.35-90)(179.35-168.69) } ; ; T = sqrt{ 13544699.82 } = 3680.31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3680.31 }{ 100 } = 73.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3680.31 }{ 90 } = 81.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3680.31 }{ 168.69 } = 43.63 ; ;

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{ 90**2+168.69**2-100**2 }{ 2 * 90 * 168.69 } ) = 29° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+168.69**2-90**2 }{ 2 * 100 * 168.69 } ) = 25° 52'12" ; ; gamma = 180° - alpha - beta = 180° - 29° - 25° 52'12" = 125° 7'48" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3680.31 }{ 179.35 } = 20.52 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 29° } = 103.13 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 168.69**2 - 100**2 } }{ 2 } = 125.614 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 168.69**2+2 * 100**2 - 90**2 } }{ 2 } = 131.164 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 100**2 - 168.69**2 } }{ 2 } = 43.995 ; ;
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