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?

1. Input data entered: side a, b and angle β.

a = 39.7 ; ; b = 26.8 ; ; beta = 33.2° ; ;

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 26.8**2 = 39.7**2 + c**2 - 2 * 39.7 * c * cos(33° 12') ; ; ; ; ; ; c**2 -66.439c +857.85 =0 ; ; a=1; b=-66.439; c=857.85 ; ; D = b**2 - 4ac = 66.439**2 - 4 * 1 * 857.85 = 982.752213445 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 66.44 ± sqrt{ 982.75 } }{ 2 } ; ; c_{1,2} = 33.21954324 ± 15.6744394911 ; ; c_{1} = 48.8939827311 ; ; c_{2} = 17.5451037489 ; ;
 ; ; (c -48.8939827311) (c -17.5451037489) = 0 ; ; ; ; c > 0 ; ; ; ; c = 48.894 ; ;


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 ; ;

3. 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 ; ;

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

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{ 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" ; ;

8. Inradius

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

9. 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?

1. Input data entered: side a, b and angle β.

a = 39.7 ; ; b = 26.8 ; ; beta = 33.2° ; ; : Nr. 1

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 26.8**2 = 39.7**2 + c**2 - 2 * 39.7 * c * cos(33° 12') ; ; ; ; ; ; c**2 -66.439c +857.85 =0 ; ; a=1; b=-66.439; c=857.85 ; ; D = b**2 - 4ac = 66.439**2 - 4 * 1 * 857.85 = 982.752213445 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 66.44 ± sqrt{ 982.75 } }{ 2 } ; ; c_{1,2} = 33.21954324 ± 15.6744394911 ; ; c_{1} = 48.8939827311 ; ; c_{2} = 17.5451037489 ; ; : Nr. 1
 ; ; (c -48.8939827311) (c -17.5451037489) = 0 ; ; ; ; c > 0 ; ; ; ; c = 48.894 ; ; : 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 = 39.7 ; ; b = 26.8 ; ; c = 48.89 ; ;

3. 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 ; ;

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

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{ 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" ; ;

8. Inradius

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

9. Circumradius

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




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