14 20 27 triangle

Obtuse scalene triangle.

Sides: a = 14   b = 20   c = 27

Area: T = 135.9944255393
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 30.2443927956° = 30°14'38″ = 0.52878561216 rad
Angle ∠ B = β = 46.01770368696° = 46°1'1″ = 0.80331488054 rad
Angle ∠ C = γ = 103.7399035174° = 103°44'21″ = 1.81105877266 rad

Height: ha = 19.42877507705
Height: hb = 13.59994255393
Height: hc = 10.07436485477

Median: ma = 22.70546250795
Median: mb = 19.03994327647
Median: mc = 10.75987173957

Inradius: r = 4.45988280457
Circumradius: R = 13.89876458567

Vertex coordinates: A[27; 0] B[0; 0] C[9.72222222222; 10.07436485477]
Centroid: CG[12.24107407407; 3.35878828492]
Coordinates of the circumscribed circle: U[13.5; -3.3010690891]
Coordinates of the inscribed circle: I[10.5; 4.45988280457]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.7566072044° = 149°45'22″ = 0.52878561216 rad
∠ B' = β' = 133.983296313° = 133°58'59″ = 0.80331488054 rad
∠ C' = γ' = 76.26109648256° = 76°15'39″ = 1.81105877266 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 = 14 ; ; b = 20 ; ; c = 27 ; ;

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

p = a+b+c = 14+20+27 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-14)(30.5-20)(30.5-27) } ; ; T = sqrt{ 18494.44 } = 135.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 135.99 }{ 14 } = 19.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 135.99 }{ 20 } = 13.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 135.99 }{ 27 } = 10.07 ; ;

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{ 14**2-20**2-27**2 }{ 2 * 20 * 27 } ) = 30° 14'38" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-14**2-27**2 }{ 2 * 14 * 27 } ) = 46° 1'1" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-14**2-20**2 }{ 2 * 20 * 14 } ) = 103° 44'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 135.99 }{ 30.5 } = 4.46 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 30° 14'38" } = 13.9 ; ;




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