14 21 27 triangle

Obtuse scalene triangle.

Sides: a = 14   b = 21   c = 27

Area: T = 145.1989531303
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 30.80659715435° = 30°48'21″ = 0.53876656327 rad
Angle ∠ B = β = 50.19223972126° = 50°11'33″ = 0.87660225908 rad
Angle ∠ C = γ = 99.00216312439° = 99°6″ = 1.72879044301 rad

Height: ha = 20.74113616147
Height: hb = 13.82875744098
Height: hc = 10.75547800965

Median: ma = 23.15216738056
Median: mb = 18.76883243791
Median: mc = 11.67326175299

Inradius: r = 4.68435332678
Circumradius: R = 13.66883408383

Vertex coordinates: A[27; 0] B[0; 0] C[8.9632962963; 10.75547800965]
Centroid: CG[11.9887654321; 3.58549266988]
Coordinates of the circumscribed circle: U[13.5; -2.13985839407]
Coordinates of the inscribed circle: I[10; 4.68435332678]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.1944028457° = 149°11'39″ = 0.53876656327 rad
∠ B' = β' = 129.8087602787° = 129°48'27″ = 0.87660225908 rad
∠ C' = γ' = 80.99883687561° = 80°59'54″ = 1.72879044301 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 = 21 ; ; c = 27 ; ;

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

p = a+b+c = 14+21+27 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-14)(31-21)(31-27) } ; ; T = sqrt{ 21080 } = 145.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 145.19 }{ 14 } = 20.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 145.19 }{ 21 } = 13.83 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 145.19 }{ 27 } = 10.75 ; ;

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**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 30° 48'21" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-14**2-27**2 }{ 2 * 14 * 27 } ) = 50° 11'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-14**2-21**2 }{ 2 * 21 * 14 } ) = 99° 6" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 145.19 }{ 31 } = 4.68 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 30° 48'21" } = 13.67 ; ;




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