18 21 27 triangle

Acute scalene triangle.

Sides: a = 18   b = 21   c = 27

Area: T = 188.7865592671
Perimeter: p = 66
Semiperimeter: s = 33

Angle ∠ A = α = 41.7522205202° = 41°45'8″ = 0.72987134507 rad
Angle ∠ B = β = 50.97771974348° = 50°58'38″ = 0.89897199387 rad
Angle ∠ C = γ = 87.27105973632° = 87°16'14″ = 1.52331592642 rad

Height: ha = 20.97661769634
Height: hb = 17.98795802543
Height: hc = 13.98441179756

Median: ma = 22.45499443206
Median: mb = 20.40222057631
Median: mc = 14.15109716981

Inradius: r = 5.72107755355
Circumradius: R = 13.51553322026

Vertex coordinates: A[27; 0] B[0; 0] C[11.33333333333; 13.98441179756]
Centroid: CG[12.77877777778; 4.66113726585]
Coordinates of the circumscribed circle: U[13.5; 0.64435872477]
Coordinates of the inscribed circle: I[12; 5.72107755355]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.2487794798° = 138°14'52″ = 0.72987134507 rad
∠ B' = β' = 129.0232802565° = 129°1'22″ = 0.89897199387 rad
∠ C' = γ' = 92.72994026368° = 92°43'46″ = 1.52331592642 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 = 21 ; ; c = 27 ; ;

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

p = a+b+c = 18+21+27 = 66 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66 }{ 2 } = 33 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33 * (33-18)(33-21)(33-27) } ; ; T = sqrt{ 35640 } = 188.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 188.79 }{ 18 } = 20.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 188.79 }{ 21 } = 17.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 188.79 }{ 27 } = 13.98 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 18**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 41° 45'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-18**2-27**2 }{ 2 * 18 * 27 } ) = 50° 58'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-18**2-21**2 }{ 2 * 21 * 18 } ) = 87° 16'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 188.79 }{ 33 } = 5.72 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 41° 45'8" } = 13.52 ; ;




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