Triangle calculator

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

You have entered side b, c and angle β.

Triangle has two solutions: a=100.7321515488; b=123; c=165 and a=120.0821584611; b=123; c=165.

#1 Obtuse scalene triangle.

Sides: a = 100.7321515488   b = 123   c = 165

Area: T = 6175.794362101
Perimeter: p = 388.7321515488
Semiperimeter: s = 194.3665757744

Angle ∠ A = α = 37.48985213058° = 37°29'19″ = 0.65442981285 rad
Angle ∠ B = β = 48° = 0.8387758041 rad
Angle ∠ C = γ = 94.51114786942° = 94°30'41″ = 1.65495364841 rad

Height: ha = 122.6198896204
Height: hb = 100.4199408472
Height: hc = 74.85881044971

Median: ma = 136.529944901
Median: mb = 122.0810584477
Median: mc = 76.36553658815

Inradius: r = 31.77440824963
Circumradius: R = 82.75664128708

Vertex coordinates: A[165; 0] B[0; 0] C[67.40325400376; 74.85881044971]
Centroid: CG[77.46875133459; 24.9532701499]
Coordinates of the circumscribed circle: U[82.5; -6.51095215831]
Coordinates of the inscribed circle: I[71.36657577438; 31.77440824963]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.5111478694° = 142°30'41″ = 0.65442981285 rad
∠ B' = β' = 132° = 0.8387758041 rad
∠ C' = γ' = 85.48985213058° = 85°29'19″ = 1.65495364841 rad




How did we calculate this triangle?

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

b = 123 ; ; c = 165 ; ; beta = 48° ; ;

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 123**2 = 165**2 + a**2 - 2 * 165 * a * cos(48° ) ; ; ; ; ; ; a**2 -220.813a +12096 =0 ; ; a=1; b=-220.813; c=12096 ; ; D = b**2 - 4ac = 220.813**2 - 4 * 1 * 12096 = 374.425175076 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 220.81 ± sqrt{ 374.43 } }{ 2 } ; ; a_{1,2} = 110.40655005 ± 9.67503456165 ; ; a_{1} = 120.081584612 ; ; a_{2} = 100.731515488 ; ;
 ; ; (a -120.081584612) (a -100.731515488) = 0 ; ; ; ; a > 0 ; ; ; ; a = 120.082 ; ;


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 = 100.73 ; ; b = 123 ; ; c = 165 ; ;

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

p = a+b+c = 100.73+123+165 = 388.73 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 388.73 }{ 2 } = 194.37 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 194.37 * (194.37-100.73)(194.37-123)(194.37-165) } ; ; T = sqrt{ 38140426.85 } = 6175.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6175.79 }{ 100.73 } = 122.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6175.79 }{ 123 } = 100.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6175.79 }{ 165 } = 74.86 ; ;

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{ 100.73**2-123**2-165**2 }{ 2 * 123 * 165 } ) = 37° 29'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 123**2-100.73**2-165**2 }{ 2 * 100.73 * 165 } ) = 48° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 165**2-100.73**2-123**2 }{ 2 * 123 * 100.73 } ) = 94° 30'41" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6175.79 }{ 194.37 } = 31.77 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100.73 }{ 2 * sin 37° 29'19" } = 82.76 ; ;





#2 Acute scalene triangle.

Sides: a = 120.0821584611   b = 123   c = 165

Area: T = 7362.136567969
Perimeter: p = 408.0821584611
Semiperimeter: s = 204.0410792305

Angle ∠ A = α = 46.51114786942° = 46°30'41″ = 0.81217784432 rad
Angle ∠ B = β = 48° = 0.8387758041 rad
Angle ∠ C = γ = 85.48985213058° = 85°29'19″ = 1.49220561694 rad

Height: ha = 122.6198896204
Height: hb = 119.7109523247
Height: hc = 89.23880082387

Median: ma = 132.5659810121
Median: mb = 130.5387517524
Median: mc = 89.2643897973

Inradius: r = 36.08216854145
Circumradius: R = 82.75664128708

Vertex coordinates: A[165; 0] B[0; 0] C[80.35502635232; 89.23880082387]
Centroid: CG[81.78334211744; 29.74660027462]
Coordinates of the circumscribed circle: U[82.5; 6.51095215831]
Coordinates of the inscribed circle: I[81.04107923054; 36.08216854145]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.4898521306° = 133°29'19″ = 0.81217784432 rad
∠ B' = β' = 132° = 0.8387758041 rad
∠ C' = γ' = 94.51114786942° = 94°30'41″ = 1.49220561694 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 123 ; ; c = 165 ; ; beta = 48° ; ; : Nr. 1

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 123**2 = 165**2 + a**2 - 2 * 165 * a * cos(48° ) ; ; ; ; ; ; a**2 -220.813a +12096 =0 ; ; a=1; b=-220.813; c=12096 ; ; D = b**2 - 4ac = 220.813**2 - 4 * 1 * 12096 = 374.425175076 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 220.81 ± sqrt{ 374.43 } }{ 2 } ; ; a_{1,2} = 110.40655005 ± 9.67503456165 ; ; a_{1} = 120.081584612 ; ; a_{2} = 100.731515488 ; ; : Nr. 1
 ; ; (a -120.081584612) (a -100.731515488) = 0 ; ; ; ; a > 0 ; ; ; ; a = 120.082 ; ; : 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 = 120.08 ; ; b = 123 ; ; c = 165 ; ;

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

p = a+b+c = 120.08+123+165 = 408.08 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 408.08 }{ 2 } = 204.04 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 204.04 * (204.04-120.08)(204.04-123)(204.04-165) } ; ; T = sqrt{ 54201041.77 } = 7362.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7362.14 }{ 120.08 } = 122.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7362.14 }{ 123 } = 119.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7362.14 }{ 165 } = 89.24 ; ;

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{ 120.08**2-123**2-165**2 }{ 2 * 123 * 165 } ) = 46° 30'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 123**2-120.08**2-165**2 }{ 2 * 120.08 * 165 } ) = 48° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 165**2-120.08**2-123**2 }{ 2 * 123 * 120.08 } ) = 85° 29'19" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7362.14 }{ 204.04 } = 36.08 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 120.08 }{ 2 * sin 46° 30'41" } = 82.76 ; ;




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