Triangle calculator SSA

Please enter two sides and a non-included angle
°


Triangle has two solutions with side c=46.96598413094 and with side c=1.34215718248

#1 Obtuse scalene triangle.

Sides: a = 32   b = 31   c = 46.96598413094

Area: T = 492.9354846256
Perimeter: p = 109.9659841309
Semiperimeter: s = 54.98799206547

Angle ∠ A = α = 42.62769572784° = 42°37'37″ = 0.74439807546 rad
Angle ∠ B = β = 41° = 0.71655849933 rad
Angle ∠ C = γ = 96.37330427216° = 96°22'23″ = 1.68220269057 rad

Height: ha = 30.8088427891
Height: hb = 31.80222481455
Height: hc = 20.99438889277

Median: ma = 36.43295669464
Median: mb = 37.07224068264
Median: mc = 21.00546024968

Inradius: r = 8.96657249481
Circumradius: R = 23.62659228439

Vertex coordinates: A[46.96598413094; 0] B[0; 0] C[24.15107065671; 20.99438889277]
Centroid: CG[23.70435159588; 6.99879629759]
Coordinates of the circumscribed circle: U[23.48799206547; -2.62325095379]
Coordinates of the inscribed circle: I[23.98799206547; 8.96657249481]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.3733042722° = 137°22'23″ = 0.74439807546 rad
∠ B' = β' = 139° = 0.71655849933 rad
∠ C' = γ' = 83.62769572784° = 83°37'37″ = 1.68220269057 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 32 ; ; b = 31 ; ; beta = 41° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 31**2 = 32**2 + c**2 -2 * 32 * c * cos (41° ) ; ; ; ; c**2 -48.301c +63 =0 ; ; p=1; q=-48.301; r=63 ; ; D = q**2 - 4pr = 48.301**2 - 4 * 1 * 63 = 2081.02651077 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 48.3 ± sqrt{ 2081.03 } }{ 2 } ; ; c_{1,2} = 24.15070657 ± 22.8091347423 ; ; c_{1} = 46.9598413123 ; ;
c_{2} = 1.34157182772 ; ; ; ; (c -46.9598413123) (c -1.34157182772) = 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 = 32 ; ; b = 31 ; ; c = 46.96 ; ;

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

p = a+b+c = 32+31+46.96 = 109.96 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 109.96 }{ 2 } = 54.98 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 54.98 * (54.98-32)(54.98-31)(54.98-46.96) } ; ; T = sqrt{ 242984.76 } = 492.93 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 492.93 }{ 32 } = 30.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 492.93 }{ 31 } = 31.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 492.93 }{ 46.96 } = 20.99 ; ;

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{ 32**2-31**2-46.96**2 }{ 2 * 31 * 46.96 } ) = 42° 37'37" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 31**2-32**2-46.96**2 }{ 2 * 32 * 46.96 } ) = 41° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 46.96**2-32**2-31**2 }{ 2 * 31 * 32 } ) = 96° 22'23" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 492.93 }{ 54.98 } = 8.97 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 32 }{ 2 * sin 42° 37'37" } = 23.63 ; ;





#2 Obtuse scalene triangle.

Sides: a = 32   b = 31   c = 1.34215718248

Area: T = 14.08224049397
Perimeter: p = 64.34215718248
Semiperimeter: s = 32.17107859124

Angle ∠ A = α = 137.3733042722° = 137°22'23″ = 2.3987611899 rad
Angle ∠ B = β = 41° = 0.71655849933 rad
Angle ∠ C = γ = 1.62769572784° = 1°37'37″ = 0.02883957613 rad

Height: ha = 0.88801503087
Height: hb = 0.90985422542
Height: hc = 20.99438889277

Median: ma = 15.01333243314
Median: mb = 16.51221139616
Median: mc = 31.49768259712

Inradius: r = 0.43877389156
Circumradius: R = 23.62659228439

Vertex coordinates: A[1.34215718248; 0] B[0; 0] C[24.15107065671; 20.99438889277]
Centroid: CG[8.49774261307; 6.99879629759]
Coordinates of the circumscribed circle: U[0.67107859124; 23.61663984656]
Coordinates of the inscribed circle: I[1.17107859124; 0.43877389156]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 42.62769572784° = 42°37'37″ = 2.3987611899 rad
∠ B' = β' = 139° = 0.71655849933 rad
∠ C' = γ' = 178.3733042722° = 178°22'23″ = 0.02883957613 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 32 ; ; b = 31 ; ; beta = 41° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 31**2 = 32**2 + c**2 -2 * 32 * c * cos (41° ) ; ; ; ; c**2 -48.301c +63 =0 ; ; p=1; q=-48.301; r=63 ; ; D = q**2 - 4pr = 48.301**2 - 4 * 1 * 63 = 2081.02651077 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 48.3 ± sqrt{ 2081.03 } }{ 2 } ; ; c_{1,2} = 24.15070657 ± 22.8091347423 ; ; c_{1} = 46.9598413123 ; ; : Nr. 1
c_{2} = 1.34157182772 ; ; ; ; (c -46.9598413123) (c -1.34157182772) = 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 = 32 ; ; b = 31 ; ; c = 1.34 ; ;

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

p = a+b+c = 32+31+1.34 = 64.34 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 64.34 }{ 2 } = 32.17 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.17 * (32.17-32)(32.17-31)(32.17-1.34) } ; ; T = sqrt{ 198.31 } = 14.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14.08 }{ 32 } = 0.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14.08 }{ 31 } = 0.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14.08 }{ 1.34 } = 20.99 ; ;

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{ 32**2-31**2-1.34**2 }{ 2 * 31 * 1.34 } ) = 137° 22'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 31**2-32**2-1.34**2 }{ 2 * 32 * 1.34 } ) = 41° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.34**2-32**2-31**2 }{ 2 * 31 * 32 } ) = 1° 37'37" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14.08 }{ 32.17 } = 0.44 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 32 }{ 2 * sin 137° 22'23" } = 23.63 ; ;




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

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