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?

1. Use Law of Cosines

a = 60 ; ; b = 50 ; ; beta = 32° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 50**2 = 60**2 + c**2 -2 * 60 * c * cos (32° ) ; ; ; ; c**2 -101.766c +1100 =0 ; ; p=1; q=-101.766; 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} = 50.88288577 ± 38.5884446981 ; ; c_{1} = 89.4713304681 ; ; c_{2} = 12.2944410719 ; ; ; ; text{ Factored form: } ; ; (c -89.4713304681) (c -12.2944410719) = 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 = 89.47 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 50**2+89.47**2-60**2 }{ 2 * 50 * 89.47 } ) = 39° 29'13" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 60**2+89.47**2-50**2 }{ 2 * 60 * 89.47 } ) = 32° ; ; gamma = 180° - alpha - beta = 180° - 39° 29'13" - 32° = 108° 30'47" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 89.47**2 - 60**2 } }{ 2 } = 65.974 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 89.47**2+2 * 60**2 - 50**2 } }{ 2 } = 71.955 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 60**2 - 89.47**2 } }{ 2 } = 32.384 ; ;







#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 - 2ac cos beta ; ; 50**2 = 60**2 + c**2 -2 * 60 * c * cos (32° ) ; ; ; ; c**2 -101.766c +1100 =0 ; ; p=1; q=-101.766; 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} = 50.88288577 ± 38.5884446981 ; ; c_{1} = 89.4713304681 ; ; c_{2} = 12.2944410719 ; ; ; ; text{ Factored form: } ; ; (c -89.4713304681) (c -12.2944410719) = 0 ; ; ; ; c>0 ; ; : Nr. 1


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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 50**2+12.29**2-60**2 }{ 2 * 50 * 12.29 } ) = 140° 30'47" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 60**2+12.29**2-50**2 }{ 2 * 60 * 12.29 } ) = 32° ; ; gamma = 180° - alpha - beta = 180° - 140° 30'47" - 32° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 12.29**2 - 60**2 } }{ 2 } = 20.63 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.29**2+2 * 60**2 - 50**2 } }{ 2 } = 35.363 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 60**2 - 12.29**2 } }{ 2 } = 54.884 ; ;
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