Triangle calculator SSA

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Triangle has two solutions with side c=221.5165817126 and with side c=66.61437533269

#1 Acute scalene triangle.

Sides: a = 234   b = 200   c = 221.5165817126

Area: T = 20423.15109807
Perimeter: p = 655.5165817126
Semiperimeter: s = 327.7587908563

Angle ∠ A = α = 67.21661851352° = 67°12'58″ = 1.17331437412 rad
Angle ∠ B = β = 52° = 0.9087571211 rad
Angle ∠ C = γ = 60.78438148648° = 60°47'2″ = 1.06108777013 rad

Height: ha = 174.5576845989
Height: hb = 204.2321509807
Height: hc = 184.3954516344

Median: ma = 175.6299236229
Median: mb = 204.726574
Median: mc = 187.3788455781

Inradius: r = 62.31216954531
Circumradius: R = 126.9021821507

Vertex coordinates: A[221.5165817126; 0] B[0; 0] C[144.0654785226; 184.3954516344]
Centroid: CG[121.8660200784; 61.46548387813]
Coordinates of the circumscribed circle: U[110.7587908563; 61.94215691815]
Coordinates of the inscribed circle: I[127.7587908563; 62.31216954531]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.7843814865° = 112°47'2″ = 1.17331437412 rad
∠ B' = β' = 128° = 0.9087571211 rad
∠ C' = γ' = 119.2166185135° = 119°12'58″ = 1.06108777013 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 = 234 ; ; b = 200 ; ; c = 221.52 ; ;

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

p = a+b+c = 234+200+221.52 = 655.52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 655.52 }{ 2 } = 327.76 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 327.76 * (327.76-234)(327.76-200)(327.76-221.52) } ; ; T = sqrt{ 417105095.98 } = 20423.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 20423.15 }{ 234 } = 174.56 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 20423.15 }{ 200 } = 204.23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20423.15 }{ 221.52 } = 184.39 ; ;

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{ 234**2-200**2-221.52**2 }{ 2 * 200 * 221.52 } ) = 67° 12'58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 200**2-234**2-221.52**2 }{ 2 * 234 * 221.52 } ) = 52° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 221.52**2-234**2-200**2 }{ 2 * 200 * 234 } ) = 60° 47'2" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20423.15 }{ 327.76 } = 62.31 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 234 }{ 2 * sin 67° 12'58" } = 126.9 ; ;





#2 Obtuse scalene triangle.

Sides: a = 234   b = 200   c = 66.61437533269

Area: T = 6141.605541328
Perimeter: p = 500.6143753327
Semiperimeter: s = 250.3076876663

Angle ∠ A = α = 112.7843814865° = 112°47'2″ = 1.96884489124 rad
Angle ∠ B = β = 52° = 0.9087571211 rad
Angle ∠ C = γ = 15.21661851352° = 15°12'58″ = 0.26655725302 rad

Height: ha = 52.49223539597
Height: hb = 61.41660541328
Height: hc = 184.3954516344

Median: ma = 92.35663536859
Median: mb = 139.9888199739
Median: mc = 215.1011492247

Inradius: r = 24.53663031777
Circumradius: R = 126.9021821507

Vertex coordinates: A[66.61437533269; 0] B[0; 0] C[144.0654785226; 184.3954516344]
Centroid: CG[70.22661795177; 61.46548387813]
Coordinates of the circumscribed circle: U[33.30768766634; 122.4532947162]
Coordinates of the inscribed circle: I[50.30768766634; 24.53663031777]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 67.21661851352° = 67°12'58″ = 1.96884489124 rad
∠ B' = β' = 128° = 0.9087571211 rad
∠ C' = γ' = 164.7843814865° = 164°47'2″ = 0.26655725302 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 234 ; ; b = 200 ; ; beta = 52° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 200**2 = 234**2 + c**2 -2 * 200 * c * cos (52° ) ; ; ; ; c**2 -288.13c +14756 =0 ; ; p=1; q=-288.129570452; r=14756 ; ; D = q**2 - 4pr = 288.13**2 - 4 * 1 * 14756 = 23994.6493691 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 288.13 ± sqrt{ 23994.65 } }{ 2 } ; ; c_{1,2} = 144.064785226 ± 77.4510318993 ; ;
c_{1} = 221.515817126 ; ; c_{2} = 66.6137533269 ; ; ; ; (c -221.515817126) (c -66.6137533269) = 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 = 234 ; ; b = 200 ; ; c = 66.61 ; ;

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

p = a+b+c = 234+200+66.61 = 500.61 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 500.61 }{ 2 } = 250.31 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 250.31 * (250.31-234)(250.31-200)(250.31-66.61) } ; ; T = sqrt{ 37719317.05 } = 6141.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6141.61 }{ 234 } = 52.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6141.61 }{ 200 } = 61.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6141.61 }{ 66.61 } = 184.39 ; ;

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{ 234**2-200**2-66.61**2 }{ 2 * 200 * 66.61 } ) = 112° 47'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 200**2-234**2-66.61**2 }{ 2 * 234 * 66.61 } ) = 52° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 66.61**2-234**2-200**2 }{ 2 * 200 * 234 } ) = 15° 12'58" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6141.61 }{ 250.31 } = 24.54 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 234 }{ 2 * sin 112° 47'2" } = 126.9 ; ;




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