Triangle calculator SSA

Triangle has two solutions with side c=51.74440868506 and with side c=11.36992217953

#1 Obtuse scalene triangle.

Sides: a = 37.5 b = 28.6 c = 51.74440868506

Area: T = 524.1422038681
Perimeter: p = 117.8444086851
Semiperimeter: s = 58.92220434253

Angle ∠ A = α = 45.1011398243° = 45°6'5″ = 0.78771678966 rad
Angle ∠ B = β = 32.7° = 32°42' = 0.57107226654 rad
Angle ∠ C = γ = 102.1998601757° = 102°11'55″ = 1.78437020916 rad

Height: h_a = 27.9544242063
Height: h_b = 36.65332894183
Height: h_c = 20.25990120179

Median: m_a = 37.36549938044
Median: m_b = 42.86444405306
Median: m_c = 21.04114440807

Inradius: r = 8.89655169952
Circumradius: R = 26.47697014606

Vertex coordinates: A[51.74440868506; 0] B[0; 0] C[31.55766543229; 20.25990120179]
Centroid: CG[27.76769137245; 6.7533004006]
Coordinates of the circumscribed circle: U[25.87220434253; -5.59330728952]
Coordinates of the inscribed circle: I[30.32220434253; 8.89655169952]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.8998601757° = 134°53'55″ = 0.78771678966 rad
∠ B' = β' = 147.3° = 147°18' = 0.57107226654 rad
∠ C' = γ' = 77.8011398243° = 77°48'5″ = 1.78437020916 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.

$a = 37.5 ; ; b = 28.6 ; ; c = 51.74 ; ;$

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

$p = a+b+c = 37.5+28.6+51.74 = 117.84 ; ;$

2. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 117.84 }{ 2 } = 58.92 ; ;$

3. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 58.92 * (58.92-37.5)(58.92-28.6)(58.92-51.74) } ; ; T = sqrt{ 274724.88 } = 524.14 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 524.14 }{ 37.5 } = 27.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 524.14 }{ 28.6 } = 36.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 524.14 }{ 51.74 } = 20.26 ; ;$

5. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 37.5**2-28.6**2-51.74**2 }{ 2 * 28.6 * 51.74 } ) = 45° 6'5" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28.6**2-37.5**2-51.74**2 }{ 2 * 37.5 * 51.74 } ) = 32° 42' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 51.74**2-37.5**2-28.6**2 }{ 2 * 28.6 * 37.5 } ) = 102° 11'55" ; ;$

6. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 524.14 }{ 58.92 } = 8.9 ; ;$

7. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37.5 }{ 2 * sin 45° 6'5" } = 26.47 ; ;$

#2 Obtuse scalene triangle.

Sides: a = 37.5 b = 28.6 c = 11.36992217953

Area: T = 115.1654600493
Perimeter: p = 77.46992217953
Semiperimeter: s = 38.73546108977

Angle ∠ A = α = 134.8998601757° = 134°53'55″ = 2.3544424757 rad
Angle ∠ B = β = 32.7° = 32°42' = 0.57107226654 rad
Angle ∠ C = γ = 12.4011398243° = 12°24'5″ = 0.21664452312 rad

Height: h_a = 6.14221120263
Height: h_b = 8.05334685659
Height: h_c = 20.25990120179

Median: m_a = 11.0477493024
Median: m_b = 23.73331962052
Median: m_c = 32.86601612738

Inradius: r = 2.97331704495
Circumradius: R = 26.47697014606

Vertex coordinates: A[11.36992217953; 0] B[0; 0] C[31.55766543229; 20.25990120179]
Centroid: CG[14.30986253728; 6.7533004006]
Coordinates of the circumscribed circle: U[5.68546108977; 25.85220849131]
Coordinates of the inscribed circle: I[10.13546108977; 2.97331704495]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 45.1011398243° = 45°6'5″ = 2.3544424757 rad
∠ B' = β' = 147.3° = 147°18' = 0.57107226654 rad
∠ C' = γ' = 167.5998601757° = 167°35'55″ = 0.21664452312 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

$a = 37.5 ; ; b = 28.6 ; ; beta = 32° 42' ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 28.6**2 = 37.5**2 + c**2 -2 * 28.6 * c * cos (32° 42') ; ; ; ; c**2 -63.113c +588.29 =0 ; ; p=1; q=-63.1133086459; r=588.29 ; ; D = q**2 - 4pr = 63.113**2 - 4 * 1 * 588.29 = 1630.12972823 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 63.11 ± sqrt{ 1630.13 } }{ 2 } ; ; c_{1,2} = 31.5566543229 ± 20.1874325276 ; ;$
$c_{1} = 51.7440868506 ; ; c_{2} = 11.3692217953 ; ; ; ; (c -51.7440868506) (c -11.3692217953) = 0 ; ; ; ; c>0 ; ;$

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.

$a = 37.5 ; ; b = 28.6 ; ; c = 11.37 ; ;$

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

$p = a+b+c = 37.5+28.6+11.37 = 77.47 ; ;$

3. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 77.47 }{ 2 } = 38.73 ; ;$

4. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.73 * (38.73-37.5)(38.73-28.6)(38.73-11.37) } ; ; T = sqrt{ 13262.89 } = 115.16 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 115.16 }{ 37.5 } = 6.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 115.16 }{ 28.6 } = 8.05 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 115.16 }{ 11.37 } = 20.26 ; ;$

6. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 37.5**2-28.6**2-11.37**2 }{ 2 * 28.6 * 11.37 } ) = 134° 53'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28.6**2-37.5**2-11.37**2 }{ 2 * 37.5 * 11.37 } ) = 32° 42' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.37**2-37.5**2-28.6**2 }{ 2 * 28.6 * 37.5 } ) = 12° 24'5" ; ;$

7. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 115.16 }{ 38.73 } = 2.97 ; ;$

8. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 37.5 }{ 2 * sin 134° 53'55" } = 26.47 ; ;$

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

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