Triangle calculator

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

You have entered side a, b and angle β.

Right scalene triangle.

Sides: a = 29.625   b = 57   c = 48.69766053745

Area: T = 721.318846711
Perimeter: p = 135.3221605374
Semiperimeter: s = 67.66108026873

Angle ∠ A = α = 31.31546010425° = 31°18'53″ = 0.54765428921 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 58.68553989575° = 58°41'7″ = 1.02442534347 rad

Height: ha = 48.69766053745
Height: hb = 25.30994198986
Height: hc = 29.625

Median: ma = 50.98996024665
Median: mb = 28.5
Median: mc = 38.34768443128

Inradius: r = 10.66108026873
Circumradius: R = 28.5

Vertex coordinates: A[48.69766053745; 0] B[0; 0] C[-0; 29.625]
Centroid: CG[16.23222017915; 9.875]
Coordinates of the circumscribed circle: U[24.34883026873; 14.81325]
Coordinates of the inscribed circle: I[10.66108026873; 10.66108026873]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.6855398958° = 148°41'7″ = 0.54765428921 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 121.3154601042° = 121°18'53″ = 1.02442534347 rad

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How did we calculate this triangle?

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

a = 29.625 ; ; b = 57 ; ; beta = 90° ; ;

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 ; ; ; ; 57**2 = 29.625**2 + c**2 - 2 * 29.625 * c * cos(90° ) ; ; ; ; ; ; c**2 -2371.359 =0 ; ; a=1; b=-0; c=-2371.359 ; ; D = b**2 - 4ac = 0**2 - 4 * 1 * (-2371.359) = 9485.4375 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ ± sqrt{ 9485.44 } }{ 2 } ; ; c_{1,2} = ± 48.6966053745 ; ; c_{1} = 48.6966053745 ; ; c_{2} = -48.6966053745 ; ; ; ; (c -48.6966053745) (c +48.6966053745) = 0 ; ;
 ; ; c > 0 ; ; ; ; c = 48.697 ; ;


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 = 29.63 ; ; b = 57 ; ; c = 48.7 ; ;

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

p = a+b+c = 29.63+57+48.7 = 135.32 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 135.32 }{ 2 } = 67.66 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 67.66 * (67.66-29.63)(67.66-57)(67.66-48.7) } ; ; T = sqrt{ 520300.33 } = 721.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 721.32 }{ 29.63 } = 48.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 721.32 }{ 57 } = 25.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 721.32 }{ 48.7 } = 29.63 ; ;

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{ 29.63**2-57**2-48.7**2 }{ 2 * 57 * 48.7 } ) = 31° 18'53" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 57**2-29.63**2-48.7**2 }{ 2 * 29.63 * 48.7 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 48.7**2-29.63**2-57**2 }{ 2 * 57 * 29.63 } ) = 58° 41'7" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 721.32 }{ 67.66 } = 10.66 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 29.63 }{ 2 * sin 31° 18'53" } = 28.5 ; ;




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