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 = 20.02110037192   b = 7.5   c = 18.56331514006

Area: T = 69.61218177523
Perimeter: p = 46.08441551198
Semiperimeter: s = 23.04220775599

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

Height: ha = 6.95438789093
Height: hb = 18.56331514006
Height: hc = 7.5

Median: ma = 10.01105018596
Median: mb = 18.9388138502
Median: mc = 11.93330485409

Inradius: r = 3.02110738407
Circumradius: R = 10.01105018596

Vertex coordinates: A[18.56331514006; 0] B[0; 0] C[18.56331514006; 7.5]
Centroid: CG[12.37554342671; 2.5]
Coordinates of the circumscribed circle: U[9.28215757003; 3.75]
Coordinates of the inscribed circle: I[15.54220775599; 3.02110738407]

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 = 7.5 ; ; alpha = 90° ; ; beta = 22° ; ;

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

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

3. From angle α, angle β and side b we calculate 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 = 7.5 * fraction{ sin(90° ) }{ sin (22° ) } = 20.02 ; ;

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{ 7.5**2+20.02**2 - 2 * 7.5 * 20.02 * cos(68° ) } ; ; c = 18.56 ; ;


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 = 20.02 ; ; b = 7.5 ; ; c = 18.56 ; ;

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

p = a+b+c = 20.02+7.5+18.56 = 46.08 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46.08 }{ 2 } = 23.04 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.04 * (23.04-20.02)(23.04-7.5)(23.04-18.56) } ; ; T = sqrt{ 4845.81 } = 69.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 69.61 }{ 20.02 } = 6.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 69.61 }{ 7.5 } = 18.56 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 69.61 }{ 18.56 } = 7.5 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 20.02**2-7.5**2-18.56**2 }{ 2 * 7.5 * 18.56 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.5**2-20.02**2-18.56**2 }{ 2 * 20.02 * 18.56 } ) = 22° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 18.56**2-20.02**2-7.5**2 }{ 2 * 7.5 * 20.02 } ) = 68° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 69.61 }{ 23.04 } = 3.02 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20.02 }{ 2 * sin 90° } = 10.01 ; ;




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