Triangle calculator

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

You have entered side a, angle β and angle γ.

Right scalene triangle.

Sides: a = 39   b = 22.51766604984   c = 45.03333209968

Area: T = 439.0754879719
Perimeter: p = 106.5549981495
Semiperimeter: s = 53.27549907476

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

Height: ha = 22.51766604984
Height: hb = 39
Height: hc = 19.5

Median: ma = 29.78767420172
Median: mb = 40.59224869896
Median: mc = 22.51766604984

Inradius: r = 8.24216697508
Circumradius: R = 22.51766604984

Vertex coordinates: A[45.03333209968; 0] B[0; 0] C[33.77549907476; 19.5]
Centroid: CG[26.26994372481; 6.5]
Coordinates of the circumscribed circle: U[22.51766604984; 0]
Coordinates of the inscribed circle: I[30.75883302492; 8.24216697508]

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

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How did we calculate this triangle?

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

a = 39 ; ; beta = 30° ; ; gamma = 90° ; ;

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

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

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

 fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 39 * fraction{ sin 30° }{ sin 60° } = 22.52 ; ;

4. From angle γ, angle α and side a we calculate c - By using the law of sines, we calculate unknown side c:

 fraction{ c }{ a } = fraction{ sin gamma }{ sin alpha } ; ; ; ; c = a * fraction{ sin gamma }{ sin alpha } ; ; ; ; c = 39 * fraction{ sin 90° }{ sin 60° } = 45.03 ; ;


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 = 39 ; ; b = 22.52 ; ; c = 45.03 ; ;

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

p = a+b+c = 39+22.52+45.03 = 106.55 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 106.55 }{ 2 } = 53.27 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 53.27 * (53.27-39)(53.27-22.52)(53.27-45.03) } ; ; T = sqrt{ 192786.75 } = 439.07 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 439.07 }{ 39 } = 22.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 439.07 }{ 22.52 } = 39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 439.07 }{ 45.03 } = 19.5 ; ;

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{ 22.52**2+45.03**2-39**2 }{ 2 * 22.52 * 45.03 } ) = 60° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 39**2+45.03**2-22.52**2 }{ 2 * 39 * 45.03 } ) = 30° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 39**2+22.52**2-45.03**2 }{ 2 * 39 * 22.52 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 439.07 }{ 53.27 } = 8.24 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39 }{ 2 * sin 60° } = 22.52 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.52**2+2 * 45.03**2 - 39**2 } }{ 2 } = 29.787 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 45.03**2+2 * 39**2 - 22.52**2 } }{ 2 } = 40.592 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.52**2+2 * 39**2 - 45.03**2 } }{ 2 } = 22.517 ; ;
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