Triangle calculator SSA

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Triangle has two solutions with side c=80.42766808977 and with side c=25.86220643894

#1 Obtuse scalene triangle.

Sides: a = 62   b = 42   c = 80.42766808977

Area: T = 1284.107688993
Perimeter: p = 184.4276680898
Semiperimeter: s = 92.21333404488

Angle ∠ A = α = 49.49901496922° = 49°29'25″ = 0.86437660594 rad
Angle ∠ B = β = 31° = 0.54110520681 rad
Angle ∠ C = γ = 99.51098503078° = 99°30'35″ = 1.7376774526 rad

Height: ha = 41.42328029009
Height: hb = 61.14879471395
Height: hc = 31.93223606444

Median: ma = 56.17113939662
Median: mb = 68.66774995912
Median: mc = 34.45112300208

Inradius: r = 13.92553917457
Circumradius: R = 40.77436845546

Vertex coordinates: A[80.42766808977; 0] B[0; 0] C[53.14443726435; 31.93223606444]
Centroid: CG[44.52436845137; 10.64441202148]
Coordinates of the circumscribed circle: U[40.21333404488; -6.73765126071]
Coordinates of the inscribed circle: I[50.21333404488; 13.92553917457]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.5109850308° = 130°30'35″ = 0.86437660594 rad
∠ B' = β' = 149° = 0.54110520681 rad
∠ C' = γ' = 80.49901496922° = 80°29'25″ = 1.7376774526 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 62 ; ; b = 42 ; ; beta = 31° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 42**2 = 62**2 + c**2 -2 * 62 * c * cos (31° ) ; ; ; ; c**2 -106.289c +2080 =0 ; ; p=1; q=-106.289; r=2080 ; ; D = q**2 - 4pr = 106.289**2 - 4 * 1 * 2080 = 2977.2973747 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 106.29 ± sqrt{ 2977.3 } }{ 2 } ; ; c_{1,2} = 53.14437264 ± 27.2823082542 ; ; c_{1} = 80.4266808942 ; ;
c_{2} = 25.8620643858 ; ; ; ; text{ Factored form: } ; ; (c -80.4266808942) (c -25.8620643858) = 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 = 62 ; ; b = 42 ; ; c = 80.43 ; ;

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

p = a+b+c = 62+42+80.43 = 184.43 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 184.43 }{ 2 } = 92.21 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 92.21 * (92.21-62)(92.21-42)(92.21-80.43) } ; ; T = sqrt{ 1648930.5 } = 1284.11 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1284.11 }{ 62 } = 41.42 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1284.11 }{ 42 } = 61.15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1284.11 }{ 80.43 } = 31.93 ; ;

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{ 42**2+80.43**2-62**2 }{ 2 * 42 * 80.43 } ) = 49° 29'25" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 62**2+80.43**2-42**2 }{ 2 * 62 * 80.43 } ) = 31° ; ; gamma = 180° - alpha - beta = 180° - 49° 29'25" - 31° = 99° 30'35" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1284.11 }{ 92.21 } = 13.93 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 62 }{ 2 * sin 49° 29'25" } = 40.77 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 80.43**2 - 62**2 } }{ 2 } = 56.171 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 80.43**2+2 * 62**2 - 42**2 } }{ 2 } = 68.667 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 62**2 - 80.43**2 } }{ 2 } = 34.451 ; ;







#2 Obtuse scalene triangle.

Sides: a = 62   b = 42   c = 25.86220643894

Area: T = 412.9188383545
Perimeter: p = 129.8622064389
Semiperimeter: s = 64.93110321947

Angle ∠ A = α = 130.5109850308° = 130°30'35″ = 2.27878265942 rad
Angle ∠ B = β = 31° = 0.54110520681 rad
Angle ∠ C = γ = 18.49901496922° = 18°29'25″ = 0.32327139913 rad

Height: ha = 13.32199478563
Height: hb = 19.66327801688
Height: hc = 31.93223606444

Median: ma = 15.98219644362
Median: mb = 42.60877831768
Median: mc = 51.3549668026

Inradius: r = 6.35993380482
Circumradius: R = 40.77436845546

Vertex coordinates: A[25.86220643894; 0] B[0; 0] C[53.14443726435; 31.93223606444]
Centroid: CG[26.3355479011; 10.64441202148]
Coordinates of the circumscribed circle: U[12.93110321947; 38.66988732515]
Coordinates of the inscribed circle: I[22.93110321947; 6.35993380482]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 49.49901496922° = 49°29'25″ = 2.27878265942 rad
∠ B' = β' = 149° = 0.54110520681 rad
∠ C' = γ' = 161.5109850308° = 161°30'35″ = 0.32327139913 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 62 ; ; b = 42 ; ; beta = 31° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 42**2 = 62**2 + c**2 -2 * 62 * c * cos (31° ) ; ; ; ; c**2 -106.289c +2080 =0 ; ; p=1; q=-106.289; r=2080 ; ; D = q**2 - 4pr = 106.289**2 - 4 * 1 * 2080 = 2977.2973747 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 106.29 ± sqrt{ 2977.3 } }{ 2 } ; ; c_{1,2} = 53.14437264 ± 27.2823082542 ; ; c_{1} = 80.4266808942 ; ; : Nr. 1
c_{2} = 25.8620643858 ; ; ; ; text{ Factored form: } ; ; (c -80.4266808942) (c -25.8620643858) = 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 = 62 ; ; b = 42 ; ; c = 25.86 ; ;

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

p = a+b+c = 62+42+25.86 = 129.86 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 129.86 }{ 2 } = 64.93 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 64.93 * (64.93-62)(64.93-42)(64.93-25.86) } ; ; T = sqrt{ 170501.59 } = 412.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 412.92 }{ 62 } = 13.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 412.92 }{ 42 } = 19.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 412.92 }{ 25.86 } = 31.93 ; ;

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{ 42**2+25.86**2-62**2 }{ 2 * 42 * 25.86 } ) = 130° 30'35" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 62**2+25.86**2-42**2 }{ 2 * 62 * 25.86 } ) = 31° ; ; gamma = 180° - alpha - beta = 180° - 130° 30'35" - 31° = 18° 29'25" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 412.92 }{ 64.93 } = 6.36 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 62 }{ 2 * sin 130° 30'35" } = 40.77 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 25.86**2 - 62**2 } }{ 2 } = 15.982 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 25.86**2+2 * 62**2 - 42**2 } }{ 2 } = 42.608 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 62**2 - 25.86**2 } }{ 2 } = 51.35 ; ;
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