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.

Right scalene triangle.

Sides: a = 16.01768029753   b = 6   c = 14.85105211205

Area: T = 44.55215633615
Perimeter: p = 36.86773240958
Semiperimeter: s = 18.43436620479

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 68° = 1.18768238914 rad

Height: ha = 5.56331031274
Height: hb = 14.85105211205
Height: hc = 6

Median: ma = 8.00884014877
Median: mb = 15.15105108016
Median: mc = 9.54664388328

Inradius: r = 2.41768590726
Circumradius: R = 8.00884014877

Vertex coordinates: A[14.85105211205; 0] B[0; 0] C[14.85105211205; 6]
Centroid: CG[9.99003474137; 2]
Coordinates of the circumscribed circle: U[7.42552605602; 3]
Coordinates of the inscribed circle: I[12.43436620479; 2.41768590726]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 112° = 1.18768238914 rad

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How did we calculate this triangle?

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

b = 6 ; ; alpha = 90° ; ; beta = 22° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 22 ° = 68 ° ; ;

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 = 6 * fraction{ sin 90° }{ sin 22° } = 16.02 ; ;

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{ 6**2+16.02**2 - 2 * 6 * 16.02 * cos 68° } ; ; c = 14.85 ; ;


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 = 16.02 ; ; b = 6 ; ; c = 14.85 ; ;

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

p = a+b+c = 16.02+6+14.85 = 36.87 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 36.87 }{ 2 } = 18.43 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.43 * (18.43-16.02)(18.43-6)(18.43-14.85) } ; ; T = sqrt{ 1984.84 } = 44.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 44.55 }{ 16.02 } = 5.56 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 44.55 }{ 6 } = 14.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 44.55 }{ 14.85 } = 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{ 6**2+14.85**2-16.02**2 }{ 2 * 6 * 14.85 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 16.02**2+14.85**2-6**2 }{ 2 * 16.02 * 14.85 } ) = 22° ; ; gamma = 180° - alpha - beta = 180° - 90° - 22° = 68° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 44.55 }{ 18.43 } = 2.42 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 16.02 }{ 2 * sin 90° } = 8.01 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6**2+2 * 14.85**2 - 16.02**2 } }{ 2 } = 8.008 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.85**2+2 * 16.02**2 - 6**2 } }{ 2 } = 15.151 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6**2+2 * 16.02**2 - 14.85**2 } }{ 2 } = 9.546 ; ;
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