Triangle calculator SSA

Please enter two sides and a non-included angle
°


Triangle has two solutions with side c=9.46992358985 and with side c=6.62111889399

#1 Acute scalene triangle.

Sides: a = 8.74   b = 3.7   c = 9.46992358985

Area: T = 16.16986732488
Perimeter: p = 21.90992358985
Semiperimeter: s = 10.95546179492

Angle ∠ A = α = 67.36442549049° = 67°21'51″ = 1.17657280462 rad
Angle ∠ B = β = 23° = 0.4011425728 rad
Angle ∠ C = γ = 89.63657450951° = 89°38'9″ = 1.56444388794 rad

Height: ha = 3.76999252285
Height: hb = 8.74398233777
Height: hc = 3.4154990063

Median: ma = 5.70880044018
Median: mb = 8.92221361932
Median: mc = 4.756627931

Inradius: r = 1.47659687032
Circumradius: R = 4.73547136307

Vertex coordinates: A[9.46992358985; 0] B[0; 0] C[8.04552124192; 3.4154990063]
Centroid: CG[5.83881494392; 1.1388330021]
Coordinates of the circumscribed circle: U[4.73546179492; 0.03301004901]
Coordinates of the inscribed circle: I[7.25546179492; 1.47659687032]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.6365745095° = 112°38'9″ = 1.17657280462 rad
∠ B' = β' = 157° = 0.4011425728 rad
∠ C' = γ' = 90.36442549049° = 90°21'51″ = 1.56444388794 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 = 8.74 ; ; b = 3.7 ; ; c = 9.47 ; ;

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

p = a+b+c = 8.74+3.7+9.47 = 21.91 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.91 }{ 2 } = 10.95 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.95 * (10.95-8.74)(10.95-3.7)(10.95-9.47) } ; ; T = sqrt{ 261.43 } = 16.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 16.17 }{ 8.74 } = 3.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 16.17 }{ 3.7 } = 8.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 16.17 }{ 9.47 } = 3.41 ; ;

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{ 8.74**2-3.7**2-9.47**2 }{ 2 * 3.7 * 9.47 } ) = 67° 21'51" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.7**2-8.74**2-9.47**2 }{ 2 * 8.74 * 9.47 } ) = 23° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.47**2-8.74**2-3.7**2 }{ 2 * 3.7 * 8.74 } ) = 89° 38'9" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 16.17 }{ 10.95 } = 1.48 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.74 }{ 2 * sin 67° 21'51" } = 4.73 ; ;





#2 Obtuse scalene triangle.

Sides: a = 8.74   b = 3.7   c = 6.62111889399

Area: T = 11.30656472174
Perimeter: p = 19.06111889399
Semiperimeter: s = 9.53105944699

Angle ∠ A = α = 112.6365745095° = 112°38'9″ = 1.96658646073 rad
Angle ∠ B = β = 23° = 0.4011425728 rad
Angle ∠ C = γ = 44.36442549049° = 44°21'51″ = 0.77443023183 rad

Height: ha = 2.58771046264
Height: hb = 6.11111606581
Height: hc = 3.4154990063

Median: ma = 3.10993683424
Median: mb = 7.52993672701
Median: mc = 5.8387701967

Inradius: r = 1.18662478519
Circumradius: R = 4.73547136307

Vertex coordinates: A[6.62111889399; 0] B[0; 0] C[8.04552124192; 3.4154990063]
Centroid: CG[4.8898800453; 1.1388330021]
Coordinates of the circumscribed circle: U[3.31105944699; 3.38548895729]
Coordinates of the inscribed circle: I[5.83105944699; 1.18662478519]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 67.36442549049° = 67°21'51″ = 1.96658646073 rad
∠ B' = β' = 157° = 0.4011425728 rad
∠ C' = γ' = 135.6365745095° = 135°38'9″ = 0.77443023183 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 8.74 ; ; b = 3.7 ; ; beta = 23° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 3.7**2 = 8.74**2 + c**2 -2 * 3.7 * c * cos (23° ) ; ; ; ; c**2 -16.09c +62.698 =0 ; ; p=1; q=-16.0904248383; r=62.6976 ; ; D = q**2 - 4pr = 16.09**2 - 4 * 1 * 62.698 = 8.11137147855 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 16.09 ± sqrt{ 8.11 } }{ 2 } ; ; c_{1,2} = 8.04521241917 ± 1.42402347931 ; ;
c_{1} = 9.46923589849 ; ; c_{2} = 6.62118893986 ; ; ; ; (c -9.46923589849) (c -6.62118893986) = 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 = 8.74 ; ; b = 3.7 ; ; c = 6.62 ; ;

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

p = a+b+c = 8.74+3.7+6.62 = 19.06 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 19.06 }{ 2 } = 9.53 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.53 * (9.53-8.74)(9.53-3.7)(9.53-6.62) } ; ; T = sqrt{ 127.82 } = 11.31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11.31 }{ 8.74 } = 2.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11.31 }{ 3.7 } = 6.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11.31 }{ 6.62 } = 3.41 ; ;

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{ 8.74**2-3.7**2-6.62**2 }{ 2 * 3.7 * 6.62 } ) = 112° 38'9" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.7**2-8.74**2-6.62**2 }{ 2 * 8.74 * 6.62 } ) = 23° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.62**2-8.74**2-3.7**2 }{ 2 * 3.7 * 8.74 } ) = 44° 21'51" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11.31 }{ 9.53 } = 1.19 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.74 }{ 2 * sin 112° 38'9" } = 4.73 ; ;




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

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