18 27 30 triangle

Acute scalene triangle.

Sides: a = 18   b = 27   c = 30

Area: T = 239.9710701337
Perimeter: p = 75
Semiperimeter: s = 37.5

Angle ∠ A = α = 36.33660575146° = 36°20'10″ = 0.63441838408 rad
Angle ∠ B = β = 62.7220387264° = 62°43'13″ = 1.09546772659 rad
Angle ∠ C = γ = 80.94435552214° = 80°56'37″ = 1.41327315469 rad

Height: ha = 26.66334112596
Height: hb = 17.77656075064
Height: hc = 15.99880467558

Median: ma = 27.08332051279
Median: mb = 20.73304124416
Median: mc = 17.36437553542

Inradius: r = 6.39992187023
Circumradius: R = 15.18993542824

Vertex coordinates: A[30; 0] B[0; 0] C[8.25; 15.99880467558]
Centroid: CG[12.75; 5.33326822519]
Coordinates of the circumscribed circle: U[15; 2.39109168778]
Coordinates of the inscribed circle: I[10.5; 6.39992187023]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.6643942485° = 143°39'50″ = 0.63441838408 rad
∠ B' = β' = 117.2879612736° = 117°16'47″ = 1.09546772659 rad
∠ C' = γ' = 99.05664447786° = 99°3'23″ = 1.41327315469 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 = 27 ; ; c = 30 ; ;

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

p = a+b+c = 18+27+30 = 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)(37.5-27)(37.5-30) } ; ; T = sqrt{ 57585.94 } = 239.97 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 239.97 }{ 18 } = 26.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 239.97 }{ 27 } = 17.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 239.97 }{ 30 } = 16 ; ;

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**2-27**2-30**2 }{ 2 * 27 * 30 } ) = 36° 20'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-18**2-30**2 }{ 2 * 18 * 30 } ) = 62° 43'13" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-18**2-27**2 }{ 2 * 27 * 18 } ) = 80° 56'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 239.97 }{ 37.5 } = 6.4 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 36° 20'10" } = 15.19 ; ;




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