12 18 27 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 18   c = 27

Area: T = 86.06106617451
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 20.74219164807° = 20°44'31″ = 0.36220147358 rad
Angle ∠ B = β = 32.08991838633° = 32°5'21″ = 0.56600619127 rad
Angle ∠ C = γ = 127.1698899656° = 127°10'8″ = 2.22195160051 rad

Height: ha = 14.34334436242
Height: hb = 9.56222957495
Height: hc = 6.3754863833

Median: ma = 22.14772345904
Median: mb = 18.8554707635
Median: mc = 7.1943747285

Inradius: r = 3.02196723419
Circumradius: R = 16.9421538334

Vertex coordinates: A[27; 0] B[0; 0] C[10.16766666667; 6.3754863833]
Centroid: CG[12.38988888889; 2.1254954611]
Coordinates of the circumscribed circle: U[13.5; -10.23655127434]
Coordinates of the inscribed circle: I[10.5; 3.02196723419]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.2588083519° = 159°15'29″ = 0.36220147358 rad
∠ B' = β' = 147.9110816137° = 147°54'39″ = 0.56600619127 rad
∠ C' = γ' = 52.8311100344° = 52°49'52″ = 2.22195160051 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 = 12 ; ; b = 18 ; ; c = 27 ; ;

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

p = a+b+c = 12+18+27 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-12)(28.5-18)(28.5-27) } ; ; T = sqrt{ 7406.44 } = 86.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 86.06 }{ 12 } = 14.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 86.06 }{ 18 } = 9.56 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 86.06 }{ 27 } = 6.37 ; ;

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{ 12**2-18**2-27**2 }{ 2 * 18 * 27 } ) = 20° 44'31" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-12**2-27**2 }{ 2 * 12 * 27 } ) = 32° 5'21" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-12**2-18**2 }{ 2 * 18 * 12 } ) = 127° 10'8" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 86.06 }{ 28.5 } = 3.02 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 20° 44'31" } = 16.94 ; ;




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