Triangle calculator

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

You have entered side a, c and angle γ.

Triangle has two solutions: a=39; b=6.24882111848; c=34 and a=39; b=58.41767194745; c=34.

#1 Obtuse scalene triangle.

Sides: a = 39   b = 6.24882111848   c = 34

Area: T = 68.13221294016
Perimeter: p = 79.24882111848
Semiperimeter: s = 39.62441055924

Angle ∠ A = α = 140.102167872° = 140°6'6″ = 2.44552355812 rad
Angle ∠ B = β = 5.89883212804° = 5°53'54″ = 0.10329451267 rad
Angle ∠ C = γ = 34° = 0.59334119457 rad

Height: ha = 3.49439553539
Height: hb = 21.80985232354
Height: hc = 4.00877723177

Median: ma = 14.74400838364
Median: mb = 36.45218856062
Median: mc = 22.15989727087

Inradius: r = 1.71994616354
Circumradius: R = 30.40109580495

Vertex coordinates: A[34; 0] B[0; 0] C[38.79435273087; 4.00877723177]
Centroid: CG[24.26545091029; 1.33659241059]
Coordinates of the circumscribed circle: U[17; 25.20435364647]
Coordinates of the inscribed circle: I[33.37658944076; 1.71994616354]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 39.89883212804° = 39°53'54″ = 2.44552355812 rad
∠ B' = β' = 174.102167872° = 174°6'6″ = 0.10329451267 rad
∠ C' = γ' = 146° = 0.59334119457 rad




How did we calculate this triangle?

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

a = 39 ; ; c = 34 ; ; gamma = 34° ; ;

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

c**2 = a**2 + b**2 - 2a b cos gamma ; ; ; ; 34**2 = 39**2 + b**2 - 2 * 39 * b * cos(34° ) ; ; ; ; ; ; b**2 -64.665b +365 =0 ; ; a=1; b=-64.665; c=365 ; ; D = b**2 - 4ac = 64.665**2 - 4 * 1 * 365 = 2721.55325717 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 64.66 ± sqrt{ 2721.55 } }{ 2 } ; ; b_{1,2} = 32.33246533 ± 26.0842541448 ; ; b_{1} = 58.4167194748 ; ; b_{2} = 6.24821118516 ; ; ; ;
 text{ Factored form: } ; ; (b -58.4167194748) (b -6.24821118516) = 0 ; ; ; ; b > 0 ; ; ; ; b = 58.417 ; ;


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 ; ; b = 6.25 ; ; c = 34 ; ;

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

p = a+b+c = 39+6.25+34 = 79.25 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 79.25 }{ 2 } = 39.62 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.62 * (39.62-39)(39.62-6.25)(39.62-34) } ; ; T = sqrt{ 4641.99 } = 68.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 68.13 }{ 39 } = 3.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 68.13 }{ 6.25 } = 21.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 68.13 }{ 34 } = 4.01 ; ;

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{ 6.25**2+34**2-39**2 }{ 2 * 6.25 * 34 } ) = 140° 6'6" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 39**2+34**2-6.25**2 }{ 2 * 39 * 34 } ) = 5° 53'54" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 39**2+6.25**2-34**2 }{ 2 * 39 * 6.25 } ) = 34° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 68.13 }{ 39.62 } = 1.72 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39 }{ 2 * sin 140° 6'6" } = 30.4 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.25**2+2 * 34**2 - 39**2 } }{ 2 } = 14.74 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 39**2 - 6.25**2 } }{ 2 } = 36.452 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.25**2+2 * 39**2 - 34**2 } }{ 2 } = 22.159 ; ;







#2 Obtuse scalene triangle.

Sides: a = 39   b = 58.41767194745   c = 34

Area: T = 636.9911191996
Perimeter: p = 131.4176719475
Semiperimeter: s = 65.70883597372

Angle ∠ A = α = 39.89883212804° = 39°53'54″ = 0.69663570724 rad
Angle ∠ B = β = 106.102167872° = 106°6'6″ = 1.85218236355 rad
Angle ∠ C = γ = 34° = 0.59334119457 rad

Height: ha = 32.66662149742
Height: hb = 21.80985232354
Height: hc = 37.47700701174

Median: ma = 43.63549235943
Median: mb = 22.03111534301
Median: mc = 46.66664393015

Inradius: r = 9.69442184304
Circumradius: R = 30.40109580495

Vertex coordinates: A[34; 0] B[0; 0] C[-10.81663693259; 37.47700701174]
Centroid: CG[7.72878768914; 12.49900233725]
Coordinates of the circumscribed circle: U[17; 25.20435364647]
Coordinates of the inscribed circle: I[7.29216402628; 9.69442184304]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.102167872° = 140°6'6″ = 0.69663570724 rad
∠ B' = β' = 73.89883212804° = 73°53'54″ = 1.85218236355 rad
∠ C' = γ' = 146° = 0.59334119457 rad

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How did we calculate this triangle?

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

a = 39 ; ; c = 34 ; ; gamma = 34° ; ; : Nr. 1

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

c**2 = a**2 + b**2 - 2a b cos gamma ; ; ; ; 34**2 = 39**2 + b**2 - 2 * 39 * b * cos(34° ) ; ; ; ; ; ; b**2 -64.665b +365 =0 ; ; a=1; b=-64.665; c=365 ; ; D = b**2 - 4ac = 64.665**2 - 4 * 1 * 365 = 2721.55325717 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 64.66 ± sqrt{ 2721.55 } }{ 2 } ; ; b_{1,2} = 32.33246533 ± 26.0842541448 ; ; b_{1} = 58.4167194748 ; ; b_{2} = 6.24821118516 ; ; ; ; : Nr. 1
 text{ Factored form: } ; ; (b -58.4167194748) (b -6.24821118516) = 0 ; ; ; ; b > 0 ; ; ; ; b = 58.417 ; ; : 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 ; ; b = 58.42 ; ; c = 34 ; ;

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

p = a+b+c = 39+58.42+34 = 131.42 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 131.42 }{ 2 } = 65.71 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 65.71 * (65.71-39)(65.71-58.42)(65.71-34) } ; ; T = sqrt{ 405757.78 } = 636.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 636.99 }{ 39 } = 32.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 636.99 }{ 58.42 } = 21.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 636.99 }{ 34 } = 37.47 ; ;

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{ 58.42**2+34**2-39**2 }{ 2 * 58.42 * 34 } ) = 39° 53'54" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 39**2+34**2-58.42**2 }{ 2 * 39 * 34 } ) = 106° 6'6" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 39**2+58.42**2-34**2 }{ 2 * 39 * 58.42 } ) = 34° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 636.99 }{ 65.71 } = 9.69 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39 }{ 2 * sin 39° 53'54" } = 30.4 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 58.42**2+2 * 34**2 - 39**2 } }{ 2 } = 43.635 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 39**2 - 58.42**2 } }{ 2 } = 22.031 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 58.42**2+2 * 39**2 - 34**2 } }{ 2 } = 46.666 ; ;
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