18 28 29 triangle

Acute scalene triangle.

Sides: a = 18   b = 28   c = 29

Area: T = 242.999884259
Perimeter: p = 75
Semiperimeter: s = 37.5

Angle ∠ A = α = 36.76439249588° = 36°45'50″ = 0.64216515365 rad
Angle ∠ B = β = 68.59659541323° = 68°35'45″ = 1.19772252532 rad
Angle ∠ C = γ = 74.64401209089° = 74°38'24″ = 1.30327158639 rad

Height: ha = 276.9998713989
Height: hb = 17.3577060185
Height: hc = 16.75985408683

Median: ma = 27.04662566726
Median: mb = 19.66596032513
Median: mc = 18.54404962177

Inradius: r = 6.48799691357
Circumradius: R = 15.03771086589

Vertex coordinates: A[29; 0] B[0; 0] C[6.56989655172; 16.75985408683]
Centroid: CG[11.85663218391; 5.58661802894]
Coordinates of the circumscribed circle: U[14.5; 3.98330436626]
Coordinates of the inscribed circle: I[9.5; 6.48799691357]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.2366075041° = 143°14'10″ = 0.64216515365 rad
∠ B' = β' = 111.4044045868° = 111°24'15″ = 1.19772252532 rad
∠ C' = γ' = 105.3659879091° = 105°21'36″ = 1.30327158639 rad

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How did we calculate this triangle?

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 = 18 ; ; b = 28 ; ; c = 29 ; ;

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

p = a+b+c = 18+28+29 = 75 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 75 }{ 2 } = 37.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37.5 * (37.5-18)(37.5-28)(37.5-29) } ; ; T = sqrt{ 59048.44 } = 243 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 243 }{ 18 } = 27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 243 }{ 28 } = 17.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 243 }{ 29 } = 16.76 ; ;

5. 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{ 28**2+29**2-18**2 }{ 2 * 28 * 29 } ) = 36° 45'50" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 18**2+29**2-28**2 }{ 2 * 18 * 29 } ) = 68° 35'45" ; ; gamma = 180° - alpha - beta = 180° - 36° 45'50" - 68° 35'45" = 74° 38'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 243 }{ 37.5 } = 6.48 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 18 }{ 2 * sin 36° 45'50" } = 15.04 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 29**2 - 18**2 } }{ 2 } = 27.046 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 29**2+2 * 18**2 - 28**2 } }{ 2 } = 19.66 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 18**2 - 29**2 } }{ 2 } = 18.54 ; ;
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