Triangle calculator - result

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

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

Acute scalene triangle.

Sides: a = 14.97325484853   b = 13   c = 11.51992819451

Area: T = 72.32440175519
Perimeter: p = 39.49218304304
Semiperimeter: s = 19.74659152152

Angle ∠ A = α = 75° = 1.3098996939 rad
Angle ∠ B = β = 57° = 0.99548376736 rad
Angle ∠ C = γ = 48° = 0.8387758041 rad

Height: ha = 9.66108827312
Height: hb = 11.12767719311
Height: hc = 12.55770357418

Median: ma = 9.73766640195
Median: mb = 11.67698557119
Median: mc = 12.78333931309

Inradius: r = 3.66327331154
Circumradius: R = 7.75503614034

Vertex coordinates: A[11.51992819451; 0] B[0; 0] C[8.15546343588; 12.55770357418]
Centroid: CG[6.55879721013; 4.18656785806]
Coordinates of the circumscribed circle: U[5.76596409725; 5.18660040254]
Coordinates of the inscribed circle: I[6.74659152152; 3.66327331154]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105° = 1.3098996939 rad
∠ B' = β' = 123° = 0.99548376736 rad
∠ C' = γ' = 132° = 0.8387758041 rad

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How did we calculate this triangle?

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

b = 13 ; ; alpha = 75° ; ; beta = 57° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 75 ° - 57 ° = 48 ° ; ;

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 = 13 * fraction{ sin 75° }{ sin 57° } = 14.97 ; ;

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{ 13**2+14.97**2 - 2 * 13 * 14.97 * cos 48° } ; ; c = 11.52 ; ;


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 = 14.97 ; ; b = 13 ; ; c = 11.52 ; ;

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

p = a+b+c = 14.97+13+11.52 = 39.49 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 39.49 }{ 2 } = 19.75 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.75 * (19.75-14.97)(19.75-13)(19.75-11.52) } ; ; T = sqrt{ 5230.76 } = 72.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 72.32 }{ 14.97 } = 9.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 72.32 }{ 13 } = 11.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 72.32 }{ 11.52 } = 12.56 ; ;

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{ 13**2+11.52**2-14.97**2 }{ 2 * 13 * 11.52 } ) = 75° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 14.97**2+11.52**2-13**2 }{ 2 * 14.97 * 11.52 } ) = 57° ; ; gamma = 180° - alpha - beta = 180° - 75° - 57° = 48° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 72.32 }{ 19.75 } = 3.66 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 14.97 }{ 2 * sin 75° } = 7.75 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 11.52**2 - 14.97**2 } }{ 2 } = 9.737 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.52**2+2 * 14.97**2 - 13**2 } }{ 2 } = 11.67 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 14.97**2 - 11.52**2 } }{ 2 } = 12.783 ; ;
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