Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side b, c and area T.

Triangle has two solutions: a=30.89331832401; b=21.8; c=10.3 and a=14.43216052224; b=21.8; c=10.3.

#1 Obtuse scalene triangle.

Sides: a = 30.89331832401   b = 21.8   c = 10.3

Area: T = 62.5
Perimeter: p = 62.99331832401
Semiperimeter: s = 31.497659162

Angle ∠ A = α = 146.1732549001° = 146°10'21″ = 2.55111922561 rad
Angle ∠ B = β = 23.13110141545° = 23°7'52″ = 0.40437123563 rad
Angle ∠ C = γ = 10.69664368444° = 10°41'47″ = 0.18766880412 rad

Height: ha = 4.04662000639
Height: hb = 5.73439449541
Height: hc = 12.13659223301

Median: ma = 7.21658026112
Median: mb = 20.2843722177
Median: mc = 26.23553175196

Inradius: r = 1.98443416949
Circumradius: R = 27.74770214589

Vertex coordinates: A[10.3; 0] B[0; 0] C[28.41096490634; 12.13659223301]
Centroid: CG[12.90332163545; 4.04553074434]
Coordinates of the circumscribed circle: U[5.15; 27.26548986765]
Coordinates of the inscribed circle: I[9.697659162; 1.98443416949]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 33.82774509989° = 33°49'39″ = 2.55111922561 rad
∠ B' = β' = 156.8698985845° = 156°52'8″ = 0.40437123563 rad
∠ C' = γ' = 169.3043563156° = 169°18'13″ = 0.18766880412 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 = 30.89 ; ; b = 21.8 ; ; c = 10.3 ; ;

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

p = a+b+c = 30.89+21.8+10.3 = 62.99 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62.99 }{ 2 } = 31.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.5 * (31.5-30.89)(31.5-21.8)(31.5-10.3) } ; ; T = sqrt{ 3906.25 } = 62.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 62.5 }{ 30.89 } = 4.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 62.5 }{ 21.8 } = 5.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 62.5 }{ 10.3 } = 12.14 ; ;

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{ 30.89**2-21.8**2-10.3**2 }{ 2 * 21.8 * 10.3 } ) = 146° 10'21" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21.8**2-30.89**2-10.3**2 }{ 2 * 30.89 * 10.3 } ) = 23° 7'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.3**2-30.89**2-21.8**2 }{ 2 * 21.8 * 30.89 } ) = 10° 41'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 62.5 }{ 31.5 } = 1.98 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 30.89 }{ 2 * sin 146° 10'21" } = 27.75 ; ;





#2 Obtuse scalene triangle.

Sides: a = 14.43216052224   b = 21.8   c = 10.3

Area: T = 62.5
Perimeter: p = 46.53216052224
Semiperimeter: s = 23.26658026112

Angle ∠ A = α = 33.82774509989° = 33°49'39″ = 0.59904003975 rad
Angle ∠ B = β = 122.7621892109° = 122°45'43″ = 2.14325992133 rad
Angle ∠ C = γ = 23.41106568924° = 23°24'38″ = 0.40985930428 rad

Height: ha = 8.66215451347
Height: hb = 5.73439449541
Height: hc = 12.13659223301

Median: ma = 15.447659162
Median: mb = 6.19444018797
Median: mc = 17.75548053959

Inradius: r = 2.68663461813
Circumradius: R = 12.96218905465

Vertex coordinates: A[10.3; 0] B[0; 0] C[-7.81096490634; 12.13659223301]
Centroid: CG[0.83301169789; 4.04553074434]
Coordinates of the circumscribed circle: U[5.15; 11.89548773235]
Coordinates of the inscribed circle: I[1.46658026112; 2.68663461813]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.1732549001° = 146°10'21″ = 0.59904003975 rad
∠ B' = β' = 57.23881078913° = 57°14'17″ = 2.14325992133 rad
∠ C' = γ' = 156.5899343108° = 156°35'22″ = 0.40985930428 rad

Calculate another triangle

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 = 14.43 ; ; b = 21.8 ; ; c = 10.3 ; ;

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

p = a+b+c = 14.43+21.8+10.3 = 46.53 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46.53 }{ 2 } = 23.27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.27 * (23.27-14.43)(23.27-21.8)(23.27-10.3) } ; ; T = sqrt{ 3906.25 } = 62.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 62.5 }{ 14.43 } = 8.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 62.5 }{ 21.8 } = 5.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 62.5 }{ 10.3 } = 12.14 ; ;

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{ 14.43**2-21.8**2-10.3**2 }{ 2 * 21.8 * 10.3 } ) = 33° 49'39" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21.8**2-14.43**2-10.3**2 }{ 2 * 14.43 * 10.3 } ) = 122° 45'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.3**2-14.43**2-21.8**2 }{ 2 * 21.8 * 14.43 } ) = 23° 24'38" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 62.5 }{ 23.27 } = 2.69 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14.43 }{ 2 * sin 33° 49'39" } = 12.96 ; ;




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