18 24 27 triangle

Acute scalene triangle.

Sides: a = 18   b = 24   c = 27

Area: T = 211.7277271508
Perimeter: p = 69
Semiperimeter: s = 34.5

Angle ∠ A = α = 40.80444376906° = 40°48'16″ = 0.71221717871 rad
Angle ∠ B = β = 60.61107200521° = 60°36'39″ = 1.05878566269 rad
Angle ∠ C = γ = 78.58548422573° = 78°35'5″ = 1.37215642395 rad

Height: ha = 23.52552523897
Height: hb = 17.64439392923
Height: hc = 15.68435015931

Median: ma = 23.90660661758
Median: mb = 19.55876072156
Median: mc = 16.3633068172

Inradius: r = 6.13770223625
Circumradius: R = 13.77224346006

Vertex coordinates: A[27; 0] B[0; 0] C[8.83333333333; 15.68435015931]
Centroid: CG[11.94444444444; 5.22878338644]
Coordinates of the circumscribed circle: U[13.5; 2.7265794348]
Coordinates of the inscribed circle: I[10.5; 6.13770223625]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.1965562309° = 139°11'44″ = 0.71221717871 rad
∠ B' = β' = 119.3899279948° = 119°23'21″ = 1.05878566269 rad
∠ C' = γ' = 101.4155157743° = 101°24'55″ = 1.37215642395 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 = 24 ; ; c = 27 ; ;

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

p = a+b+c = 18+24+27 = 69 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 69 }{ 2 } = 34.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.5 * (34.5-18)(34.5-24)(34.5-27) } ; ; T = sqrt{ 44828.44 } = 211.73 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 211.73 }{ 18 } = 23.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 211.73 }{ 24 } = 17.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 211.73 }{ 27 } = 15.68 ; ;

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**2-24**2-27**2 }{ 2 * 24 * 27 } ) = 40° 48'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-18**2-27**2 }{ 2 * 18 * 27 } ) = 60° 36'39" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-18**2-24**2 }{ 2 * 24 * 18 } ) = 78° 35'5" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 211.73 }{ 34.5 } = 6.14 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 40° 48'16" } = 13.77 ; ;




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