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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. Circumradius

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




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