Triangle calculator

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

You have entered side b, c and angle γ.

Triangle has two solutions: a=113.5244019669; b=500; c=400 and a=792.7843767367; b=500; c=400.

#1 Obtuse scalene triangle.

Sides: a = 113.5244019669   b = 500   c = 400

Area: T = 11994.33109646
Perimeter: p = 1013.524401967
Semiperimeter: s = 506.7622009835

Angle ∠ A = α = 6.88988308305° = 6°53'20″ = 0.12202327796 rad
Angle ∠ B = β = 148.1111169169° = 148°6'40″ = 2.5855027561 rad
Angle ∠ C = γ = 25° = 0.4366332313 rad

Height: ha = 211.309913087
Height: hb = 47.97773238584
Height: hc = 59.9721654823

Median: ma = 449.1977144069
Median: mb = 154.7388009296
Median: mc = 302.3976844429

Inradius: r = 23.66985677534
Circumradius: R = 473.244031663

Vertex coordinates: A[400; 0] B[0; 0] C[-96.39903711977; 59.9721654823]
Centroid: CG[101.2033209601; 19.99105516077]
Coordinates of the circumscribed circle: U[200; 428.9011384102]
Coordinates of the inscribed circle: I[6.76220098346; 23.66985677534]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 173.1111169169° = 173°6'40″ = 0.12202327796 rad
∠ B' = β' = 31.88988308305° = 31°53'20″ = 2.5855027561 rad
∠ C' = γ' = 155° = 0.4366332313 rad




How did we calculate this triangle?

1. Input data entered: side b, c and angle γ.

b = 500 ; ; c = 400 ; ; gamma = 25° ; ;

2. From angle γ, side b and c we calculate a - by using the law of cosines and quadratic equation:

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 400**2 = 500**2 + a**2 - 2 * 500 * a * cos(25° ) ; ; ; ; ; ; a**2 -906.308a +90000 =0 ; ; a=1; b=-906.308; c=90000 ; ; D = b**2 - 4ac = 906.308**2 - 4 * 1 * 90000 = 461393.804843 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 906.31 ± sqrt{ 461393.8 } }{ 2 } ; ; a_{1,2} = 453.15389352 ± 339.629873849 ; ; a_{1} = 792.783767369 ; ; a_{2} = 113.524019671 ; ;
 ; ; (a -792.783767369) (a -113.524019671) = 0 ; ; ; ; a > 0 ; ; ; ; a = 792.784 ; ;


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 = 113.52 ; ; b = 500 ; ; c = 400 ; ;

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

p = a+b+c = 113.52+500+400 = 1013.52 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1013.52 }{ 2 } = 506.76 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 506.76 * (506.76-113.52)(506.76-500)(506.76-400) } ; ; T = sqrt{ 143863975.29 } = 11994.33 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11994.33 }{ 113.52 } = 211.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11994.33 }{ 500 } = 47.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11994.33 }{ 400 } = 59.97 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 113.52**2-500**2-400**2 }{ 2 * 500 * 400 } ) = 6° 53'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 500**2-113.52**2-400**2 }{ 2 * 113.52 * 400 } ) = 148° 6'40" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 400**2-113.52**2-500**2 }{ 2 * 500 * 113.52 } ) = 25° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11994.33 }{ 506.76 } = 23.67 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 113.52 }{ 2 * sin 6° 53'20" } = 473.24 ; ;





#2 Obtuse scalene triangle.

Sides: a = 792.7843767367   b = 500   c = 400

Area: T = 83761.22444253
Perimeter: p = 1692.784376737
Semiperimeter: s = 846.3921883684

Angle ∠ A = α = 123.1111169169° = 123°6'40″ = 2.1498695248 rad
Angle ∠ B = β = 31.88988308305° = 31°53'20″ = 0.55765650926 rad
Angle ∠ C = γ = 25° = 0.4366332313 rad

Height: ha = 211.309913087
Height: hb = 335.0454897701
Height: hc = 418.8066122126

Median: ma = 218.8800078953
Median: mb = 575.9880078562
Median: mc = 631.8654740986

Inradius: r = 98.96326980598
Circumradius: R = 473.244031663

Vertex coordinates: A[400; 0] B[0; 0] C[673.1332627252; 418.8066122126]
Centroid: CG[357.7110875751; 139.6022040709]
Coordinates of the circumscribed circle: U[200; 428.9011384102]
Coordinates of the inscribed circle: I[346.3921883684; 98.96326980598]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 56.88988308305° = 56°53'20″ = 2.1498695248 rad
∠ B' = β' = 148.1111169169° = 148°6'40″ = 0.55765650926 rad
∠ C' = γ' = 155° = 0.4366332313 rad

Calculate another triangle

How did we calculate this triangle?

1. Input data entered: side b, c and angle γ.

b = 500 ; ; c = 400 ; ; gamma = 25° ; ; : Nr. 1

2. From angle γ, side b and c we calculate a - by using the law of cosines and quadratic equation:

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 400**2 = 500**2 + a**2 - 2 * 500 * a * cos(25° ) ; ; ; ; ; ; a**2 -906.308a +90000 =0 ; ; a=1; b=-906.308; c=90000 ; ; D = b**2 - 4ac = 906.308**2 - 4 * 1 * 90000 = 461393.804843 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 906.31 ± sqrt{ 461393.8 } }{ 2 } ; ; a_{1,2} = 453.15389352 ± 339.629873849 ; ; a_{1} = 792.783767369 ; ; a_{2} = 113.524019671 ; ; : Nr. 1
 ; ; (a -792.783767369) (a -113.524019671) = 0 ; ; ; ; a > 0 ; ; ; ; a = 792.784 ; ; : 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 = 792.78 ; ; b = 500 ; ; c = 400 ; ;

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

p = a+b+c = 792.78+500+400 = 1692.78 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1692.78 }{ 2 } = 846.39 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 846.39 * (846.39-792.78)(846.39-500)(846.39-400) } ; ; T = sqrt{ 7015942717.22 } = 83761.22 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 83761.22 }{ 792.78 } = 211.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 83761.22 }{ 500 } = 335.04 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 83761.22 }{ 400 } = 418.81 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 792.78**2-500**2-400**2 }{ 2 * 500 * 400 } ) = 123° 6'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 500**2-792.78**2-400**2 }{ 2 * 792.78 * 400 } ) = 31° 53'20" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 400**2-792.78**2-500**2 }{ 2 * 500 * 792.78 } ) = 25° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 83761.22 }{ 846.39 } = 98.96 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 792.78 }{ 2 * sin 123° 6'40" } = 473.24 ; ;




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