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 = 10.35504700351   b = 11.33   c = 4.6088326166

Area: T = 23.84991709468
Perimeter: p = 26.28987962011
Semiperimeter: s = 13.14443981006

Angle ∠ A = α = 66° = 1.15219173063 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 24° = 0.41988790205 rad

Height: ha = 4.6088326166
Height: hb = 4.21099154363
Height: hc = 10.35504700351

Median: ma = 6.93296267966
Median: mb = 5.665
Median: mc = 10.60438388077

Inradius: r = 1.81443981006
Circumradius: R = 5.665

Vertex coordinates: A[4.6088326166; 0] B[0; 0] C[-0; 10.35504700351]
Centroid: CG[1.5366108722; 3.45501566784]
Coordinates of the circumscribed circle: U[2.3044163083; 5.17552350175]
Coordinates of the inscribed circle: I[1.81443981006; 1.81443981006]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 114° = 1.15219173063 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 156° = 0.41988790205 rad

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How did we calculate this triangle?

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

b = 11.33 ; ; alpha = 66° ; ; beta = 90° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 66 ° - 90 ° = 24 ° ; ;

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 = 11.33 * fraction{ sin 66° }{ sin 90° } = 10.35 ; ;

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{ 11.33**2+10.35**2 - 2 * 11.33 * 10.35 * cos 24° } ; ; c = 4.61 ; ;
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 = 10.35 ; ; b = 11.33 ; ; c = 4.61 ; ;

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

p = a+b+c = 10.35+11.33+4.61 = 26.29 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.29 }{ 2 } = 13.14 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.14 * (13.14-10.35)(13.14-11.33)(13.14-4.61) } ; ; T = sqrt{ 568.78 } = 23.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 23.85 }{ 10.35 } = 4.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 23.85 }{ 11.33 } = 4.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 23.85 }{ 4.61 } = 10.35 ; ;

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{ 11.33**2+4.61**2-10.35**2 }{ 2 * 11.33 * 4.61 } ) = 66° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.35**2+4.61**2-11.33**2 }{ 2 * 10.35 * 4.61 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 66° - 90° = 24° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 23.85 }{ 13.14 } = 1.81 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10.35 }{ 2 * sin 66° } = 5.67 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.33**2+2 * 4.61**2 - 10.35**2 } }{ 2 } = 6.93 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.61**2+2 * 10.35**2 - 11.33**2 } }{ 2 } = 5.665 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.33**2+2 * 10.35**2 - 4.61**2 } }{ 2 } = 10.604 ; ;
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