Triangle calculator SSA

Please enter two sides and a non-included angle
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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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 42**2 = 62**2 + c**2 -2 * 42 * c * cos (31° ) ; ; ; ; c**2 -106.289c +2080 =0 ; ; p=1; q=-106.288745287; 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} = 53.1443726435 ± 27.2823082542 ; ; c_{1} = 80.4266808977 ; ;
c_{2} = 25.8620643894 ; ; ; ; (c -80.4266808977) (c -25.8620643894) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 62**2-42**2-25.86**2 }{ 2 * 42 * 25.86 } ) = 130° 30'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 42**2-62**2-25.86**2 }{ 2 * 62 * 25.86 } ) = 31° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25.86**2-62**2-42**2 }{ 2 * 42 * 62 } ) = 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 ; ;




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