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 = 19.03992489729   b = 18.75   c = 3.30661308883

Area: T = 30.99549770777
Perimeter: p = 41.09553798611
Semiperimeter: s = 20.54876899306

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 80° = 1.39662634016 rad
Angle ∠ C = γ = 10° = 0.17545329252 rad

Height: ha = 3.25659033313
Height: hb = 3.30661308883
Height: hc = 18.75

Median: ma = 9.52196244864
Median: mb = 9.94108815731
Median: mc = 18.82327289563

Inradius: r = 1.50884409577
Circumradius: R = 9.52196244864

Vertex coordinates: A[3.30661308883; 0] B[0; 0] C[3.30661308883; 18.75]
Centroid: CG[2.20440872589; 6.25]
Coordinates of the circumscribed circle: U[1.65330654441; 9.375]
Coordinates of the inscribed circle: I[1.79876899306; 1.50884409577]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 100° = 1.39662634016 rad
∠ C' = γ' = 170° = 0.17545329252 rad

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How did we calculate this triangle?

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

b = 18.75 ; ; alpha = 90° ; ; beta = 80° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 80 ° = 10 ° ; ;

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 = 18.75 * fraction{ sin 90° }{ sin 80° } = 19.04 ; ;

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{ 18.75**2+19.04**2 - 2 * 18.75 * 19.04 * cos 10° } ; ; c = 3.31 ; ;
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 = 19.04 ; ; b = 18.75 ; ; c = 3.31 ; ;

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

p = a+b+c = 19.04+18.75+3.31 = 41.1 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 41.1 }{ 2 } = 20.55 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.55 * (20.55-19.04)(20.55-18.75)(20.55-3.31) } ; ; T = sqrt{ 960.69 } = 30.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 30.99 }{ 19.04 } = 3.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 30.99 }{ 18.75 } = 3.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 30.99 }{ 3.31 } = 18.75 ; ;

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{ 18.75**2+3.31**2-19.04**2 }{ 2 * 18.75 * 3.31 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 19.04**2+3.31**2-18.75**2 }{ 2 * 19.04 * 3.31 } ) = 80° ; ; gamma = 180° - alpha - beta = 180° - 90° - 80° = 10° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 30.99 }{ 20.55 } = 1.51 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 19.04 }{ 2 * sin 90° } = 9.52 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.75**2+2 * 3.31**2 - 19.04**2 } }{ 2 } = 9.52 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.31**2+2 * 19.04**2 - 18.75**2 } }{ 2 } = 9.941 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.75**2+2 * 19.04**2 - 3.31**2 } }{ 2 } = 18.823 ; ;
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