Triangle calculator

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

You have entered side a, angle α and angle γ.

Acute scalene triangle.

Sides: a = 19.2   b = 20.76994277294   c = 21.63334471857

Area: T = 181.2899411096
Perimeter: p = 61.60328749151
Semiperimeter: s = 30.80114374576

Angle ∠ A = α = 53.8° = 53°48' = 0.93989871376 rad
Angle ∠ B = β = 60.8° = 60°48' = 1.06111601852 rad
Angle ∠ C = γ = 65.4° = 65°24' = 1.14114453308 rad

Height: ha = 18.88443136558
Height: hb = 17.45773332937
Height: hc = 16.76601038836

Median: ma = 18.90883997914
Median: mb = 17.62204635727
Median: mc = 16.82326946361

Inradius: r = 5.88657451489
Circumradius: R = 11.89664958444

Vertex coordinates: A[21.63334471857; 0] B[0; 0] C[9.36769054555; 16.76601038836]
Centroid: CG[10.33334508804; 5.58767012945]
Coordinates of the circumscribed circle: U[10.81767235928; 4.95222827152]
Coordinates of the inscribed circle: I[10.03220097281; 5.88657451489]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.2° = 126°12' = 0.93989871376 rad
∠ B' = β' = 119.2° = 119°12' = 1.06111601852 rad
∠ C' = γ' = 114.6° = 114°36' = 1.14114453308 rad

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How did we calculate this triangle?

1. Input data entered: side a, angle α and angle γ.

a = 19.2 ; ; alpha = 53.8° ; ; gamma = 65.4° ; ;

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

 alpha + gamma + beta = 180° ; ; beta = 180° - alpha - gamma = 180° - 53.8 ° - 65.4 ° = 60.8 ° ; ;

3. From angle β, angle α and side a we calculate b - By using the law of sines, we calculate unknown side b:

 fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 19.2 * fraction{ sin(60° 48') }{ sin (53° 48') } = 20.77 ; ;

4. From angle γ, angle α and side a we calculate c - By using the law of sines, we calculate unknown side c:

 fraction{ c }{ a } = fraction{ sin( gamma ) }{ sin ( alpha ) } ; ; ; ; c = a * fraction{ sin( gamma ) }{ sin ( alpha ) } ; ; ; ; c = 19.2 * fraction{ sin(65° 24') }{ sin (53° 48') } = 21.63 ; ;


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 = 19.2 ; ; b = 20.77 ; ; c = 21.63 ; ;

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

p = a+b+c = 19.2+20.77+21.63 = 61.6 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61.6 }{ 2 } = 30.8 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.8 * (30.8-19.2)(30.8-20.77)(30.8-21.63) } ; ; T = sqrt{ 32865.85 } = 181.29 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 181.29 }{ 19.2 } = 18.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 181.29 }{ 20.77 } = 17.46 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 181.29 }{ 21.63 } = 16.76 ; ;

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{ 19.2**2-20.77**2-21.63**2 }{ 2 * 20.77 * 21.63 } ) = 53° 48' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20.77**2-19.2**2-21.63**2 }{ 2 * 19.2 * 21.63 } ) = 60° 48' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21.63**2-19.2**2-20.77**2 }{ 2 * 20.77 * 19.2 } ) = 65° 24' ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 181.29 }{ 30.8 } = 5.89 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19.2 }{ 2 * sin 53° 48' } = 11.9 ; ;




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