Triangle calculator

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

You have entered side c, angle α, angle β and angle γ.

Acute scalene triangle.

Sides: a = 44.44774045078   b = 59.08334825345   c = 56

Area: T = 1176.724370478
Perimeter: p = 159.5310887042
Semiperimeter: s = 79.76554435211

Angle ∠ A = α = 45.34404620062° = 45°20'26″ = 0.79113403464 rad
Angle ∠ B = β = 71° = 1.23991837689 rad
Angle ∠ C = γ = 63.66595379938° = 63°39'34″ = 1.11110685383 rad

Height: ha = 52.94990402336
Height: hb = 39.83325777121
Height: hc = 42.02658465993

Median: ma = 53.09993033131
Median: mb = 41.02552532794
Median: mc = 44.15499132269

Inradius: r = 14.75222994023
Circumradius: R = 31.24439565225

Vertex coordinates: A[56; 0] B[0; 0] C[14.4710659456; 42.02658465993]
Centroid: CG[23.49902198187; 14.00986155331]
Coordinates of the circumscribed circle: U[28; 13.86330739441]
Coordinates of the inscribed circle: I[20.68219609866; 14.75222994023]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.6659537994° = 134°39'34″ = 0.79113403464 rad
∠ B' = β' = 109° = 1.23991837689 rad
∠ C' = γ' = 116.3440462006° = 116°20'26″ = 1.11110685383 rad

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How did we calculate this triangle?

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

c = 56 ; ; alpha = 46° ; ; beta = 71° ; ; gamma = 65° ; ;

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

 fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 56 * fraction{ sin(46° ) }{ sin (65° ) } = 44.45 ; ;

3. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 44.45**2+56**2 - 2 * 44.45 * 56 * cos(71° ) } ; ; b = 59.08 ; ;


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 = 44.45 ; ; b = 59.08 ; ; c = 56 ; ;

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

p = a+b+c = 44.45+59.08+56 = 159.53 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 159.53 }{ 2 } = 79.77 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 79.77 * (79.77-44.45)(79.77-59.08)(79.77-56) } ; ; T = sqrt{ 1384678.68 } = 1176.72 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1176.72 }{ 44.45 } = 52.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1176.72 }{ 59.08 } = 39.83 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1176.72 }{ 56 } = 42.03 ; ;

8. 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{ 44.45**2-59.08**2-56**2 }{ 2 * 59.08 * 56 } ) = 45° 20'26" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 59.08**2-44.45**2-56**2 }{ 2 * 44.45 * 56 } ) = 71° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 56**2-44.45**2-59.08**2 }{ 2 * 59.08 * 44.45 } ) = 63° 39'34" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1176.72 }{ 79.77 } = 14.75 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 44.45 }{ 2 * sin 45° 20'26" } = 31.24 ; ;




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