Triangle calculator SSA

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Triangle has two solutions with side c=136.3033383791 and with side c=55.2087726989

#1 Acute scalene triangle.

Sides: a = 125   b = 90   c = 136.3033383791

Area: T = 5475.883289119
Perimeter: p = 351.3033383791
Semiperimeter: s = 175.6521691895

Angle ∠ A = α = 63.22222151534° = 63°13'20″ = 1.10334358148 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 76.77877848466° = 76°46'40″ = 1.3440025138 rad

Height: ha = 87.6144126259
Height: hb = 121.6866286471
Height: hc = 80.34884512108

Median: ma = 97.12439219575
Median: mb = 122.788764684
Median: mc = 84.95879124732

Inradius: r = 31.17546663644
Circumradius: R = 70.00875722087

Vertex coordinates: A[136.3033383791; 0] B[0; 0] C[95.75655553899; 80.34884512108]
Centroid: CG[77.35329797269; 26.78328170703]
Coordinates of the circumscribed circle: U[68.15216918953; 16.01327155211]
Coordinates of the inscribed circle: I[85.65216918953; 31.17546663644]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.7787784847° = 116°46'40″ = 1.10334358148 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 103.2222215153° = 103°13'20″ = 1.3440025138 rad




How did we calculate this triangle?

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 = 125 ; ; b = 90 ; ; c = 136.3 ; ;

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

p = a+b+c = 125+90+136.3 = 351.3 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 351.3 }{ 2 } = 175.65 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 175.65 * (175.65-125)(175.65-90)(175.65-136.3) } ; ; T = sqrt{ 29985293.44 } = 5475.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5475.88 }{ 125 } = 87.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5475.88 }{ 90 } = 121.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5475.88 }{ 136.3 } = 80.35 ; ;

5. 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{ 125**2-90**2-136.3**2 }{ 2 * 90 * 136.3 } ) = 63° 13'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-125**2-136.3**2 }{ 2 * 125 * 136.3 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 136.3**2-125**2-90**2 }{ 2 * 90 * 125 } ) = 76° 46'40" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5475.88 }{ 175.65 } = 31.17 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 125 }{ 2 * sin 63° 13'20" } = 70.01 ; ;





#2 Obtuse scalene triangle.

Sides: a = 125   b = 90   c = 55.2087726989

Area: T = 2217.928767922
Perimeter: p = 270.2087726989
Semiperimeter: s = 135.1043863494

Angle ∠ A = α = 116.7787784847° = 116°46'40″ = 2.03881568388 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 23.22222151534° = 23°13'20″ = 0.4055304114 rad

Height: ha = 35.48768428675
Height: hb = 49.28772817604
Height: hc = 80.34884512108

Median: ma = 40.83774406599
Median: mb = 85.50769971385
Median: mc = 105.3599037202

Inradius: r = 16.41664637624
Circumradius: R = 70.00875722087

Vertex coordinates: A[55.2087726989; 0] B[0; 0] C[95.75655553899; 80.34884512108]
Centroid: CG[50.32110941263; 26.78328170703]
Coordinates of the circumscribed circle: U[27.60438634945; 64.33657356897]
Coordinates of the inscribed circle: I[45.10438634945; 16.41664637624]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 63.22222151534° = 63°13'20″ = 2.03881568388 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 156.7787784847° = 156°46'40″ = 0.4055304114 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 125 ; ; b = 90 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 125**2 + c**2 -2 * 90 * c * cos (40° ) ; ; ; ; c**2 -191.511c +7525 =0 ; ; p=1; q=-191.51111078; r=7525 ; ; D = q**2 - 4pr = 191.511**2 - 4 * 1 * 7525 = 6576.50555209 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 191.51 ± sqrt{ 6576.51 } }{ 2 } ; ; c_{1,2} = 95.7555553899 ± 40.5478284008 ; ; c_{1} = 136.303383791 ; ;
c_{2} = 55.207726989 ; ; ; ; (c -136.303383791) (c -55.207726989) = 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 = 125 ; ; b = 90 ; ; c = 55.21 ; ;

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

p = a+b+c = 125+90+55.21 = 270.21 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 270.21 }{ 2 } = 135.1 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 135.1 * (135.1-125)(135.1-90)(135.1-55.21) } ; ; T = sqrt{ 4919203.19 } = 2217.93 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2217.93 }{ 125 } = 35.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2217.93 }{ 90 } = 49.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2217.93 }{ 55.21 } = 80.35 ; ;

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{ 125**2-90**2-55.21**2 }{ 2 * 90 * 55.21 } ) = 116° 46'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-125**2-55.21**2 }{ 2 * 125 * 55.21 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 55.21**2-125**2-90**2 }{ 2 * 90 * 125 } ) = 23° 13'20" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2217.93 }{ 135.1 } = 16.42 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 125 }{ 2 * sin 116° 46'40" } = 70.01 ; ;




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