Triangle calculator

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

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

Right scalene triangle.

Sides: a = 43.01325950546   b = 37.25   c = 21.50662975273

Area: T = 400.5554791446
Perimeter: p = 101.7698892582
Semiperimeter: s = 50.8844446291

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 30° = 0.52435987756 rad

Height: ha = 18.625
Height: hb = 21.50662975273
Height: hc = 37.25

Median: ma = 21.50662975273
Median: mb = 28.45501574395
Median: mc = 38.77110292401

Inradius: r = 7.87218512363
Circumradius: R = 21.50662975273

Vertex coordinates: A[21.50662975273; 0] B[0; 0] C[21.50662975273; 37.25]
Centroid: CG[14.33875316849; 12.41766666667]
Coordinates of the circumscribed circle: U[10.75331487637; 18.625]
Coordinates of the inscribed circle: I[13.6344446291; 7.87218512363]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 150° = 0.52435987756 rad

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How did we calculate this triangle?

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

b = 37.25 ; ; alpha = 90° ; ; beta = 60° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 60 ° = 30 ° ; ;

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 = 37.25 * fraction{ sin 90° }{ sin 60° } = 43.01 ; ;

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{ 37.25**2+43.01**2 - 2 * 37.25 * 43.01 * cos(30° ) } ; ; c = 21.51 ; ;


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 = 43.01 ; ; b = 37.25 ; ; c = 21.51 ; ;

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

p = a+b+c = 43.01+37.25+21.51 = 101.77 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 101.77 }{ 2 } = 50.88 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 50.88 * (50.88-43.01)(50.88-37.25)(50.88-21.51) } ; ; T = sqrt{ 160444.14 } = 400.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 400.55 }{ 43.01 } = 18.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 400.55 }{ 37.25 } = 21.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 400.55 }{ 21.51 } = 37.25 ; ;

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{ 37.25**2+21.51**2-43.01**2 }{ 2 * 37.25 * 21.51 } ) = 90° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 43.01**2+21.51**2-37.25**2 }{ 2 * 43.01 * 21.51 } ) = 60° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 43.01**2+37.25**2-21.51**2 }{ 2 * 43.01 * 37.25 } ) = 30° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 400.55 }{ 50.88 } = 7.87 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 43.01 }{ 2 * sin 90° } = 21.51 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 37.25**2+2 * 21.51**2 - 43.01**2 } }{ 2 } = 21.506 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.51**2+2 * 43.01**2 - 37.25**2 } }{ 2 } = 28.45 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 37.25**2+2 * 43.01**2 - 21.51**2 } }{ 2 } = 38.771 ; ;
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