10 19 27 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 19   c = 27

Area: T = 67.35498329619
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 15.22327570342° = 15°13'22″ = 0.26656872315 rad
Angle ∠ B = β = 29.92664348666° = 29°55'35″ = 0.52223148218 rad
Angle ∠ C = γ = 134.8510808099° = 134°51'3″ = 2.35435906003 rad

Height: ha = 13.47699665924
Height: hb = 7.08994561013
Height: hc = 4.98988765157

Median: ma = 22.8043508502
Median: mb = 18.00769431054
Median: mc = 6.94662219947

Inradius: r = 2.40553511772
Circumradius: R = 19.04223634863

Vertex coordinates: A[27; 0] B[0; 0] C[8.66766666667; 4.98988765157]
Centroid: CG[11.88988888889; 1.66329588386]
Coordinates of the circumscribed circle: U[13.5; -13.43298774061]
Coordinates of the inscribed circle: I[9; 2.40553511772]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 164.7777242966° = 164°46'38″ = 0.26656872315 rad
∠ B' = β' = 150.0743565133° = 150°4'25″ = 0.52223148218 rad
∠ C' = γ' = 45.14991919008° = 45°8'57″ = 2.35435906003 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 = 10 ; ; b = 19 ; ; c = 27 ; ;

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

p = a+b+c = 10+19+27 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-10)(28-19)(28-27) } ; ; T = sqrt{ 4536 } = 67.35 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 67.35 }{ 10 } = 13.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 67.35 }{ 19 } = 7.09 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 67.35 }{ 27 } = 4.99 ; ;

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{ 10**2-19**2-27**2 }{ 2 * 19 * 27 } ) = 15° 13'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-10**2-27**2 }{ 2 * 10 * 27 } ) = 29° 55'35" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-10**2-19**2 }{ 2 * 19 * 10 } ) = 134° 51'3" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 67.35 }{ 28 } = 2.41 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 15° 13'22" } = 19.04 ; ;




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