Triangle calculator SSA

Please enter two sides and a non-included angle
°


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?

1. Use Law of Cosines

a = 234 ; ; b = 200 ; ; beta = 52° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 200**2 = 234**2 + c**2 -2 * 234 * c * cos (52° ) ; ; ; ; c**2 -288.13c +14756 =0 ; ; p=1; q=-288.13; 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} = 144.06478523 ± 77.4510318993 ; ; c_{1} = 221.515817129 ; ; c_{2} = 66.6137533307 ; ; ; ; text{ Factored form: } ; ; (c -221.515817129) (c -66.6137533307) = 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 = 221.52 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 200**2+221.52**2-234**2 }{ 2 * 200 * 221.52 } ) = 67° 12'58" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 234**2+221.52**2-200**2 }{ 2 * 234 * 221.52 } ) = 52° ; ; gamma = 180° - alpha - beta = 180° - 67° 12'58" - 52° = 60° 47'2" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 200**2+2 * 221.52**2 - 234**2 } }{ 2 } = 175.629 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 221.52**2+2 * 234**2 - 200**2 } }{ 2 } = 204.726 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 200**2+2 * 234**2 - 221.52**2 } }{ 2 } = 187.378 ; ;







#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 - 2ac cos beta ; ; 200**2 = 234**2 + c**2 -2 * 234 * c * cos (52° ) ; ; ; ; c**2 -288.13c +14756 =0 ; ; p=1; q=-288.13; 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} = 144.06478523 ± 77.4510318993 ; ; c_{1} = 221.515817129 ; ; c_{2} = 66.6137533307 ; ; ; ; text{ Factored form: } ; ; (c -221.515817129) (c -66.6137533307) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 200**2+66.61**2-234**2 }{ 2 * 200 * 66.61 } ) = 112° 47'2" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 234**2+66.61**2-200**2 }{ 2 * 234 * 66.61 } ) = 52° ; ; gamma = 180° - alpha - beta = 180° - 112° 47'2" - 52° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 200**2+2 * 66.61**2 - 234**2 } }{ 2 } = 92.356 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 66.61**2+2 * 234**2 - 200**2 } }{ 2 } = 139.988 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 200**2+2 * 234**2 - 66.61**2 } }{ 2 } = 215.101 ; ;
Calculate another triangle

Look also our friend's collection of math examples and problems:

See more informations about triangles or more information about solving triangles.