Triangle calculator

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

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

Acute scalene triangle.

Sides: a = 21.99111485751   b = 28.27112756537   c = 21.99111485751

Area: T = 238.1321741867
Perimeter: p = 72.2543572804
Semiperimeter: s = 36.1276786402

Angle ∠ A = α = 50° = 0.8732664626 rad
Angle ∠ B = β = 80° = 1.39662634016 rad
Angle ∠ C = γ = 50° = 0.8732664626 rad

Height: ha = 21.65770536144
Height: hb = 16.84661971638
Height: hc = 21.65770536144

Median: ma = 22.81552398072
Median: mb = 16.84661971638
Median: mc = 22.81552398072

Inradius: r = 6.59215561716
Circumradius: R = 14.35437028254

Vertex coordinates: A[21.99111485751; 0] B[0; 0] C[3.81987228749; 21.65770536144]
Centroid: CG[8.60332904833; 7.21990178715]
Coordinates of the circumscribed circle: U[10.99655742876; 9.22663823293]
Coordinates of the inscribed circle: I[7.85655107483; 6.59215561716]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130° = 0.8732664626 rad
∠ B' = β' = 100° = 1.39662634016 rad
∠ C' = γ' = 130° = 0.8732664626 rad

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How did we calculate this triangle?

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

a = 21.991 ; ; alpha = 50° ; ; gamma = 50° ; ;

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

 alpha + gamma + beta = 180° ; ; beta = 180° - alpha - gamma = 180° - 50 ° - 50 ° = 80 ° ; ;

3. From angle β, angle α and side a we calculate side 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 = 21.99 * fraction{ sin 80° }{ sin 50° } = 28.27 ; ;

4. From angle γ, angle α and side a we calculate side 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 = 21.99 * fraction{ sin 50° }{ sin 50° } = 21.99 ; ;
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 = 21.99 ; ; b = 28.27 ; ; c = 21.99 ; ;

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

p = a+b+c = 21.99+28.27+21.99 = 72.25 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72.25 }{ 2 } = 36.13 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36.13 * (36.13-21.99)(36.13-28.27)(36.13-21.99) } ; ; T = sqrt{ 56706.73 } = 238.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 238.13 }{ 21.99 } = 21.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 238.13 }{ 28.27 } = 16.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 238.13 }{ 21.99 } = 21.66 ; ;

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{ 28.27**2+21.99**2-21.99**2 }{ 2 * 28.27 * 21.99 } ) = 50° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 21.99**2+21.99**2-28.27**2 }{ 2 * 21.99 * 21.99 } ) = 80° ; ;
 gamma = 180° - alpha - beta = 180° - 50° - 80° = 50° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 238.13 }{ 36.13 } = 6.59 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 21.99 }{ 2 * sin 50° } = 14.35 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 28.27**2+2 * 21.99**2 - 21.99**2 } }{ 2 } = 22.815 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.99**2+2 * 21.99**2 - 28.27**2 } }{ 2 } = 16.846 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 28.27**2+2 * 21.99**2 - 21.99**2 } }{ 2 } = 22.815 ; ;
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