Triangle calculator SSA

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Triangle has two solutions with side c=89.47113304675 and with side c=12.29444410713

#1 Obtuse scalene triangle.

Sides: a = 60   b = 50   c = 89.47113304675

Area: T = 1422.377744834
Perimeter: p = 199.4711330467
Semiperimeter: s = 99.73656652338

Angle ∠ A = α = 39.4876999201° = 39°29'13″ = 0.68991781478 rad
Angle ∠ B = β = 32° = 0.55985053606 rad
Angle ∠ C = γ = 108.5133000799° = 108°30'47″ = 1.89439091452 rad

Height: ha = 47.41325816113
Height: hb = 56.89550979336
Height: hc = 31.7955155854

Median: ma = 65.97439303651
Median: mb = 71.95552603207
Median: mc = 32.38439505943

Inradius: r = 14.26114725134
Circumradius: R = 47.177699787

Vertex coordinates: A[89.47113304675; 0] B[0; 0] C[50.88328857694; 31.7955155854]
Centroid: CG[46.78547387456; 10.59883852847]
Coordinates of the circumscribed circle: U[44.73656652338; -14.98796323092]
Coordinates of the inscribed circle: I[49.73656652338; 14.26114725134]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.5133000799° = 140°30'47″ = 0.68991781478 rad
∠ B' = β' = 148° = 0.55985053606 rad
∠ C' = γ' = 71.4876999201° = 71°29'13″ = 1.89439091452 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 = 60 ; ; b = 50 ; ; c = 89.47 ; ;

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

p = a+b+c = 60+50+89.47 = 199.47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 199.47 }{ 2 } = 99.74 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 99.74 * (99.74-60)(99.74-50)(99.74-89.47) } ; ; T = sqrt{ 2023157.61 } = 1422.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1422.38 }{ 60 } = 47.41 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1422.38 }{ 50 } = 56.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1422.38 }{ 89.47 } = 31.8 ; ;

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{ 60**2-50**2-89.47**2 }{ 2 * 50 * 89.47 } ) = 39° 29'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 50**2-60**2-89.47**2 }{ 2 * 60 * 89.47 } ) = 32° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 89.47**2-60**2-50**2 }{ 2 * 50 * 60 } ) = 108° 30'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1422.38 }{ 99.74 } = 14.26 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 60 }{ 2 * sin 39° 29'13" } = 47.18 ; ;





#2 Obtuse scalene triangle.

Sides: a = 60   b = 50   c = 12.29444410713

Area: T = 195.4521834999
Perimeter: p = 122.2944441071
Semiperimeter: s = 61.14772205356

Angle ∠ A = α = 140.5133000799° = 140°30'47″ = 2.45224145058 rad
Angle ∠ B = β = 32° = 0.55985053606 rad
Angle ∠ C = γ = 7.4876999201° = 7°29'13″ = 0.13106727872 rad

Height: ha = 6.51550611666
Height: hb = 7.81880734
Height: hc = 31.7955155854

Median: ma = 20.63295089769
Median: mb = 35.3633493049
Median: mc = 54.88436194113

Inradius: r = 3.19664140526
Circumradius: R = 47.177699787

Vertex coordinates: A[12.29444410713; 0] B[0; 0] C[50.88328857694; 31.7955155854]
Centroid: CG[21.05991089469; 10.59883852847]
Coordinates of the circumscribed circle: U[6.14772205356; 46.77547881632]
Coordinates of the inscribed circle: I[11.14772205356; 3.19664140526]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 39.4876999201° = 39°29'13″ = 2.45224145058 rad
∠ B' = β' = 148° = 0.55985053606 rad
∠ C' = γ' = 172.5133000799° = 172°30'47″ = 0.13106727872 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 60 ; ; b = 50 ; ; beta = 32° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 50**2 = 60**2 + c**2 -2 * 50 * c * cos (32° ) ; ; ; ; c**2 -101.766c +1100 =0 ; ; p=1; q=-101.765771539; r=1100 ; ; D = q**2 - 4pr = 101.766**2 - 4 * 1 * 1100 = 5956.27225688 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 101.77 ± sqrt{ 5956.27 } }{ 2 } ; ; c_{1,2} = 50.8828857694 ± 38.5884446981 ; ; c_{1} = 89.4713304675 ; ;
c_{2} = 12.2944410713 ; ; ; ; (c -89.4713304675) (c -12.2944410713) = 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 = 60 ; ; b = 50 ; ; c = 12.29 ; ;

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

p = a+b+c = 60+50+12.29 = 122.29 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 122.29 }{ 2 } = 61.15 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 61.15 * (61.15-60)(61.15-50)(61.15-12.29) } ; ; T = sqrt{ 38201.42 } = 195.45 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 195.45 }{ 60 } = 6.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 195.45 }{ 50 } = 7.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 195.45 }{ 12.29 } = 31.8 ; ;

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{ 60**2-50**2-12.29**2 }{ 2 * 50 * 12.29 } ) = 140° 30'47" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 50**2-60**2-12.29**2 }{ 2 * 60 * 12.29 } ) = 32° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.29**2-60**2-50**2 }{ 2 * 50 * 60 } ) = 7° 29'13" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 195.45 }{ 61.15 } = 3.2 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 60 }{ 2 * sin 140° 30'47" } = 47.18 ; ;




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