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=14.79221109117; b=33; c=42 and a=45.63224323167; b=33; c=42.

#1 Obtuse scalene triangle.

Sides: a = 14.79221109117   b = 33   c = 42

Area: T = 215.7854736893
Perimeter: p = 89.79221109117
Semiperimeter: s = 44.89660554559

Angle ∠ A = α = 18.1422275913° = 18°8'32″ = 0.31766424485 rad
Angle ∠ B = β = 44° = 0.76879448709 rad
Angle ∠ C = γ = 117.8587724087° = 117°51'28″ = 2.05770053342 rad

Height: ha = 29.17656515593
Height: hb = 13.0787862842
Height: hc = 10.27554636616

Median: ma = 37.03877964206
Median: mb = 26.81770332552
Median: mc = 14.59112053173

Inradius: r = 4.80663183882
Circumradius: R = 23.75326829038

Vertex coordinates: A[42; 0] B[0; 0] C[10.64105541098; 10.27554636616]
Centroid: CG[17.54768513699; 3.42551545539]
Coordinates of the circumscribed circle: U[21; -11.09990965907]
Coordinates of the inscribed circle: I[11.89660554559; 4.80663183882]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.8587724087° = 161°51'28″ = 0.31766424485 rad
∠ B' = β' = 136° = 0.76879448709 rad
∠ C' = γ' = 62.1422275913° = 62°8'32″ = 2.05770053342 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 = 14.79 ; ; b = 33 ; ; c = 42 ; ;

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

p = a+b+c = 14.79+33+42 = 89.79 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 89.79 }{ 2 } = 44.9 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 44.9 * (44.9-14.79)(44.9-33)(44.9-42) } ; ; T = sqrt{ 46563.05 } = 215.78 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 215.78 }{ 14.79 } = 29.18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 215.78 }{ 33 } = 13.08 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 215.78 }{ 42 } = 10.28 ; ;

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{ 14.79**2-33**2-42**2 }{ 2 * 33 * 42 } ) = 18° 8'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 33**2-14.79**2-42**2 }{ 2 * 14.79 * 42 } ) = 44° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 42**2-14.79**2-33**2 }{ 2 * 33 * 14.79 } ) = 117° 51'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 215.78 }{ 44.9 } = 4.81 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14.79 }{ 2 * sin 18° 8'32" } = 23.75 ; ;





#2 Acute scalene triangle.

Sides: a = 45.63224323167   b = 33   c = 42

Area: T = 665.6787972538
Perimeter: p = 120.6322432317
Semiperimeter: s = 60.31662161584

Angle ∠ A = α = 73.8587724087° = 73°51'28″ = 1.28990604633 rad
Angle ∠ B = β = 44° = 0.76879448709 rad
Angle ∠ C = γ = 62.1422275913° = 62°8'32″ = 1.08545873194 rad

Height: ha = 29.17656515593
Height: hb = 40.34441195477
Height: hc = 31.69989510732

Median: ma = 30.09985096012
Median: mb = 40.63113849084
Median: mc = 33.8332816016

Inradius: r = 11.03664677186
Circumradius: R = 23.75326829038

Vertex coordinates: A[42; 0] B[0; 0] C[32.82552247517; 31.69989510732]
Centroid: CG[24.94217415839; 10.56663170244]
Coordinates of the circumscribed circle: U[21; 11.09990965907]
Coordinates of the inscribed circle: I[27.31662161584; 11.03664677186]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 106.1422275913° = 106°8'32″ = 1.28990604633 rad
∠ B' = β' = 136° = 0.76879448709 rad
∠ C' = γ' = 117.8587724087° = 117°51'28″ = 1.08545873194 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 = 45.63 ; ; b = 33 ; ; c = 42 ; ;

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

p = a+b+c = 45.63+33+42 = 120.63 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 120.63 }{ 2 } = 60.32 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 60.32 * (60.32-45.63)(60.32-33)(60.32-42) } ; ; T = sqrt{ 443127.16 } = 665.68 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 665.68 }{ 45.63 } = 29.18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 665.68 }{ 33 } = 40.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 665.68 }{ 42 } = 31.7 ; ;

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{ 45.63**2-33**2-42**2 }{ 2 * 33 * 42 } ) = 73° 51'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 33**2-45.63**2-42**2 }{ 2 * 45.63 * 42 } ) = 44° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 42**2-45.63**2-33**2 }{ 2 * 33 * 45.63 } ) = 62° 8'32" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 665.68 }{ 60.32 } = 11.04 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 45.63 }{ 2 * sin 73° 51'28" } = 23.75 ; ;




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