Triangle calculator

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

You have entered side b, angle α and angle γ.

Right scalene triangle.

Sides: a = 600.6321864937   b = 1415   c = 1537.219993403

Area: T = 424947.0444443
Perimeter: p = 3552.832179897
Semiperimeter: s = 1776.416589948

Angle ∠ A = α = 23° = 0.4011425728 rad
Angle ∠ B = β = 67° = 1.16993705988 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 1415
Height: hb = 600.6321864937
Height: hc = 552.8854546812

Median: ma = 1446.518811578
Median: mb = 928.0710518429
Median: mc = 768.6599967014

Inradius: r = 239.2165965454
Circumradius: R = 768.6599967014

Vertex coordinates: A[1537.219993403; 0] B[0; 0] C[234.6865566393; 552.8854546812]
Centroid: CG[590.6298500141; 184.2954848937]
Coordinates of the circumscribed circle: U[768.6599967014; 0]
Coordinates of the inscribed circle: I[361.4165899482; 239.2165965454]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157° = 0.4011425728 rad
∠ B' = β' = 113° = 1.16993705988 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: side b, angle α and angle γ.

b = 1415 ; ; alpha = 23° ; ; gamma = 90° ; ;

2. From angle α and angle γ we calculate angle β:

 alpha + gamma + beta = 180° ; ; beta = 180° - alpha - gamma = 180° - 23 ° - 90 ° = 67 ° ; ;

3. From angle α, angle β and side b we calculate side a - By using the law of sines, we calculate unknown side a:

 fraction{ a }{ b } = fraction{ sin alpha }{ sin beta } ; ; ; ; a = b * fraction{ sin alpha }{ sin beta } ; ; ; ; a = 1415 * fraction{ sin 23° }{ sin 67° } = 600.63 ; ;

4. Calculation of the third side c of the triangle using a Law of Cosines

c**2 = b**2+a**2 - 2ba cos gamma ; ; c = sqrt{ b**2+a**2 - 2ba cos gamma } ; ; c = sqrt{ 1415**2+600.63**2 - 2 * 1415 * 600.63 * cos 90° } ; ; c = 1537.2 ; ;
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 = 600.63 ; ; b = 1415 ; ; c = 1537.2 ; ;

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

p = a+b+c = 600.63+1415+1537.2 = 3552.83 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 3552.83 }{ 2 } = 1776.42 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1776.42 * (1776.42-600.63)(1776.42-1415)(1776.42-1537.2) } ; ; T = sqrt{ 180579990581 } = 424947.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 424947.04 }{ 600.63 } = 1415 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 424947.04 }{ 1415 } = 600.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 424947.04 }{ 1537.2 } = 552.88 ; ;

9. 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{ 1415**2+1537.2**2-600.63**2 }{ 2 * 1415 * 1537.2 } ) = 23° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 600.63**2+1537.2**2-1415**2 }{ 2 * 600.63 * 1537.2 } ) = 67° ; ; gamma = 180° - alpha - beta = 180° - 23° - 67° = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 424947.04 }{ 1776.42 } = 239.22 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 600.63 }{ 2 * sin 23° } = 768.6 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1415**2+2 * 1537.2**2 - 600.63**2 } }{ 2 } = 1446.518 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1537.2**2+2 * 600.63**2 - 1415**2 } }{ 2 } = 928.071 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1415**2+2 * 600.63**2 - 1537.2**2 } }{ 2 } = 768.6 ; ;
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