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?

1. Input data entered: side b, c and area T.

b = 21.8 ; ; c = 10.3 ; ; T = 62.5 ; ;

2. From area T, side b and side c we calculate side a - using Heron's formula for the area and solve of the bikvadratic equation:

s = fraction{ a+b+c }{ 2 } ; ; T**2 = s(s-a)(s-b)(s-c) ; ; ; ; s = fraction{ a+21.8+10.3 }{ 2 } = fraction{ a+32.1 }{ 2 } = a/2 + 16.05 ; ; ; ; T**2 = s(s-a)(s-b)(s-c) ; ; T**2 = ( a/2 + 16.05) ( a/2 + 16.05-a) ( a/2 + 16.05-21.8) ( a/2 + 16.05 - 10.3) ; ; ; ; 62.5**2 = ( a/2 + 16.05) ( 16.05-a/2) ( a/2 + (-5.75)) ( a/2 + 5.75) ; ; 62500 = ( a + 32.1) ( 32.1-a) ( a + (-11.5)) ( a + 11.5) ; ; ; ; D = b**2 * c**2 - 4 * S**2 = 21.8**2 * 10.3**2 - 4 * 62.5**2 = 34793.212 ; ; ; ; D_1 = -2 * sqrt{ D } + b**2 + c**2 = -2 * sqrt{ 34793.212 } + 21.8**2 + 10.3**2 = 208.271 ; ; D_2 = 2 * sqrt{ D } + b**2 + c**2 = 2 * sqrt{ 34793.212 } + 21.8**2 + 10.3**2 = 954.389 ; ; ; ; a_1 = sqrt{ D_1 } = sqrt{ 208.271 } = 14.432 ; ; a_2 = sqrt{ D_2 } = sqrt{ 954.389 } = 30.893 ; ;


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 ; ;

3. 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 ; ;

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

7. 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{ 21.8**2+10.3**2-30.89**2 }{ 2 * 21.8 * 10.3 } ) = 146° 10'21" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 30.89**2+10.3**2-21.8**2 }{ 2 * 30.89 * 10.3 } ) = 23° 7'52" ; ; gamma = 180° - alpha - beta = 180° - 146° 10'21" - 23° 7'52" = 10° 41'47" ; ;

8. Inradius

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

9. Circumradius

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

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.8**2+2 * 10.3**2 - 30.89**2 } }{ 2 } = 7.216 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.3**2+2 * 30.89**2 - 21.8**2 } }{ 2 } = 20.284 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.8**2+2 * 30.89**2 - 10.3**2 } }{ 2 } = 26.235 ; ;







#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?

1. Input data entered: side b, c and area T.

b = 21.8 ; ; c = 10.3 ; ; T = 62.5 ; ; : Nr. 1

2. From area T, side b and side c we calculate side a - using Heron's formula for the area and solve of the bikvadratic equation:

s = fraction{ a+b+c }{ 2 } ; ; T**2 = s(s-a)(s-b)(s-c) ; ; ; ; s = fraction{ a+21.8+10.3 }{ 2 } = fraction{ a+32.1 }{ 2 } = a/2 + 16.05 ; ; ; ; T**2 = s(s-a)(s-b)(s-c) ; ; T**2 = ( a/2 + 16.05) ( a/2 + 16.05-a) ( a/2 + 16.05-21.8) ( a/2 + 16.05 - 10.3) ; ; ; ; 62.5**2 = ( a/2 + 16.05) ( 16.05-a/2) ( a/2 + (-5.75)) ( a/2 + 5.75) ; ; 62500 = ( a + 32.1) ( 32.1-a) ( a + (-11.5)) ( a + 11.5) ; ; ; ; D = b**2 * c**2 - 4 * S**2 = 21.8**2 * 10.3**2 - 4 * 62.5**2 = 34793.212 ; ; ; ; D_1 = -2 * sqrt{ D } + b**2 + c**2 = -2 * sqrt{ 34793.212 } + 21.8**2 + 10.3**2 = 208.271 ; ; D_2 = 2 * sqrt{ D } + b**2 + c**2 = 2 * sqrt{ 34793.212 } + 21.8**2 + 10.3**2 = 954.389 ; ; ; ; a_1 = sqrt{ D_1 } = sqrt{ 208.271 } = 14.432 ; ; a_2 = sqrt{ D_2 } = sqrt{ 954.389 } = 30.893 ; ; : 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 = 14.43 ; ; b = 21.8 ; ; c = 10.3 ; ;

3. 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 ; ;

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

7. 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{ 21.8**2+10.3**2-14.43**2 }{ 2 * 21.8 * 10.3 } ) = 33° 49'39" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 14.43**2+10.3**2-21.8**2 }{ 2 * 14.43 * 10.3 } ) = 122° 45'43" ; ; gamma = 180° - alpha - beta = 180° - 33° 49'39" - 122° 45'43" = 23° 24'38" ; ;

8. Inradius

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

9. Circumradius

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

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.8**2+2 * 10.3**2 - 14.43**2 } }{ 2 } = 15.447 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.3**2+2 * 14.43**2 - 21.8**2 } }{ 2 } = 6.194 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.8**2+2 * 14.43**2 - 10.3**2 } }{ 2 } = 17.755 ; ;
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