Triangle calculator

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

You have entered side a, b and angle β.

Triangle has two solutions: a=39.7; b=26.8; c=17.54551037532 and a=39.7; b=26.8; c=48.89439827355.

#1 Obtuse scalene triangle.

Sides: a = 39.7   b = 26.8   c = 17.54551037532

Area: T = 190.7700013317
Perimeter: p = 84.04551037532
Semiperimeter: s = 42.02325518766

Angle ∠ A = α = 125.7943604984° = 125°47'37″ = 2.19655125849 rad
Angle ∠ B = β = 33.2° = 33°12' = 0.57994493117 rad
Angle ∠ C = γ = 21.00663950164° = 21°23″ = 0.3676630757 rad

Height: ha = 9.60770535676
Height: hb = 14.23113442774
Height: hc = 21.73882599727

Median: ma = 10.90993002917
Median: mb = 27.61215977961
Median: mc = 32.71440235002

Inradius: r = 4.53880398096
Circumradius: R = 24.47220598921

Vertex coordinates: A[17.54551037532; 0] B[0; 0] C[33.22195432443; 21.73882599727]
Centroid: CG[16.92215489992; 7.24660866576]
Coordinates of the circumscribed circle: U[8.77325518766; 22.84656571132]
Coordinates of the inscribed circle: I[15.22325518766; 4.53880398096]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 54.20663950164° = 54°12'23″ = 2.19655125849 rad
∠ B' = β' = 146.8° = 146°48' = 0.57994493117 rad
∠ C' = γ' = 158.9943604984° = 158°59'37″ = 0.3676630757 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 = 39.7 ; ; b = 26.8 ; ; c = 17.55 ; ;

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

p = a+b+c = 39.7+26.8+17.55 = 84.05 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 84.05 }{ 2 } = 42.02 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 42.02 * (42.02-39.7)(42.02-26.8)(42.02-17.55) } ; ; T = sqrt{ 36366.5 } = 190.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 190.7 }{ 39.7 } = 9.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 190.7 }{ 26.8 } = 14.23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 190.7 }{ 17.55 } = 21.74 ; ;

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{ 39.7**2-26.8**2-17.55**2 }{ 2 * 26.8 * 17.55 } ) = 125° 47'37" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 26.8**2-39.7**2-17.55**2 }{ 2 * 39.7 * 17.55 } ) = 33° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 17.55**2-39.7**2-26.8**2 }{ 2 * 26.8 * 39.7 } ) = 21° 23" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 190.7 }{ 42.02 } = 4.54 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.7 }{ 2 * sin 125° 47'37" } = 24.47 ; ;





#2 Obtuse scalene triangle.

Sides: a = 39.7   b = 26.8   c = 48.89439827355

Area: T = 531.4355053901
Perimeter: p = 115.3943982736
Semiperimeter: s = 57.69769913677

Angle ∠ A = α = 54.20663950164° = 54°12'23″ = 0.94660800687 rad
Angle ∠ B = β = 33.2° = 33°12' = 0.57994493117 rad
Angle ∠ C = γ = 92.59436049836° = 92°35'37″ = 1.61660632733 rad

Height: ha = 26.7732546796
Height: hb = 39.65993323807
Height: hc = 21.73882599727

Median: ma = 34.06547658713
Median: mb = 42.47111169369
Median: mc = 23.4421621383

Inradius: r = 9.21107931679
Circumradius: R = 24.47220598921

Vertex coordinates: A[48.89439827355; 0] B[0; 0] C[33.22195432443; 21.73882599727]
Centroid: CG[27.37111753266; 7.24660866576]
Coordinates of the circumscribed circle: U[24.44769913677; -1.10773971405]
Coordinates of the inscribed circle: I[30.89769913677; 9.21107931679]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.7943604984° = 125°47'37″ = 0.94660800687 rad
∠ B' = β' = 146.8° = 146°48' = 0.57994493117 rad
∠ C' = γ' = 87.40663950164° = 87°24'23″ = 1.61660632733 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 = 39.7 ; ; b = 26.8 ; ; c = 48.89 ; ;

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

p = a+b+c = 39.7+26.8+48.89 = 115.39 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 115.39 }{ 2 } = 57.7 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 57.7 * (57.7-39.7)(57.7-26.8)(57.7-48.89) } ; ; T = sqrt{ 282423.22 } = 531.44 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 531.44 }{ 39.7 } = 26.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 531.44 }{ 26.8 } = 39.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 531.44 }{ 48.89 } = 21.74 ; ;

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{ 39.7**2-26.8**2-48.89**2 }{ 2 * 26.8 * 48.89 } ) = 54° 12'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 26.8**2-39.7**2-48.89**2 }{ 2 * 39.7 * 48.89 } ) = 33° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 48.89**2-39.7**2-26.8**2 }{ 2 * 26.8 * 39.7 } ) = 92° 35'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 531.44 }{ 57.7 } = 9.21 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.7 }{ 2 * sin 54° 12'23" } = 24.47 ; ;




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