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 = 26.50330702185   b = 25.6   c = 6.85994993262

Area: T = 87.80215913758
Perimeter: p = 58.96325695447
Semiperimeter: s = 29.48112847724

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 75° = 1.3098996939 rad
Angle ∠ C = γ = 15° = 0.26217993878 rad

Height: ha = 6.62657675546
Height: hb = 6.85994993262
Height: hc = 25.6

Median: ma = 13.25215351092
Median: mb = 14.52221462259
Median: mc = 25.82987278578

Inradius: r = 2.97882145539
Circumradius: R = 13.25215351092

Vertex coordinates: A[6.85994993262; 0] B[0; 0] C[6.85994993262; 25.6]
Centroid: CG[4.57329995508; 8.53333333333]
Coordinates of the circumscribed circle: U[3.43297496631; 12.8]
Coordinates of the inscribed circle: I[3.88112847724; 2.97882145539]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 105° = 1.3098996939 rad
∠ C' = γ' = 165° = 0.26217993878 rad

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How did we calculate this triangle?

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

b = 25.6 ; ; alpha = 90° ; ; beta = 75° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 75 ° = 15 ° ; ;

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 = 25.6 * fraction{ sin 90° }{ sin 75° } = 26.5 ; ;

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{ 25.6**2+26.5**2 - 2 * 25.6 * 26.5 * cos(15° ) } ; ; c = 6.86 ; ;


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 = 26.5 ; ; b = 25.6 ; ; c = 6.86 ; ;

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

p = a+b+c = 26.5+25.6+6.86 = 58.96 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58.96 }{ 2 } = 29.48 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.48 * (29.48-26.5)(29.48-25.6)(29.48-6.86) } ; ; T = sqrt{ 7709.12 } = 87.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 87.8 }{ 26.5 } = 6.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 87.8 }{ 25.6 } = 6.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 87.8 }{ 6.86 } = 25.6 ; ;

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{ 25.6**2+6.86**2-26.5**2 }{ 2 * 25.6 * 6.86 } ) = 90° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 26.5**2+6.86**2-25.6**2 }{ 2 * 26.5 * 6.86 } ) = 75° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 26.5**2+25.6**2-6.86**2 }{ 2 * 26.5 * 25.6 } ) = 15° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 87.8 }{ 29.48 } = 2.98 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 26.5 }{ 2 * sin 90° } = 13.25 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 25.6**2+2 * 6.86**2 - 26.5**2 } }{ 2 } = 13.252 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.86**2+2 * 26.5**2 - 25.6**2 } }{ 2 } = 14.522 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 25.6**2+2 * 26.5**2 - 6.86**2 } }{ 2 } = 25.829 ; ;
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