17 21 27 triangle

Acute scalene triangle.

Sides: a = 17   b = 21   c = 27

Area: T = 178.549982493
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 39.02327570225° = 39°1'22″ = 0.68110755932 rad
Angle ∠ B = β = 51.05774891935° = 51°3'27″ = 0.89111212942 rad
Angle ∠ C = γ = 89.9219753784° = 89°55'11″ = 1.56993957661 rad

Height: ha = 210.9999794035
Height: hb = 176.9999833267
Height: hc = 13.22222092541

Median: ma = 22.64439837484
Median: mb = 19.96987255477
Median: mc = 13.51985058346

Inradius: r = 5.49223023055
Circumradius: R = 13.55000132406

Vertex coordinates: A[27; 0] B[0; 0] C[10.68551851852; 13.22222092541]
Centroid: CG[12.56217283951; 4.40774030847]
Coordinates of the circumscribed circle: U[13.5; 0.01989075816]
Coordinates of the inscribed circle: I[11.5; 5.49223023055]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.9777242977° = 140°58'38″ = 0.68110755932 rad
∠ B' = β' = 128.9432510806° = 128°56'33″ = 0.89111212942 rad
∠ C' = γ' = 90.0880246216° = 90°4'49″ = 1.56993957661 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 = 17 ; ; b = 21 ; ; c = 27 ; ;

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

p = a+b+c = 17+21+27 = 65 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 65 }{ 2 } = 32.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.5 * (32.5-17)(32.5-21)(32.5-27) } ; ; T = sqrt{ 31862.19 } = 178.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 178.5 }{ 17 } = 21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 178.5 }{ 21 } = 17 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 178.5 }{ 27 } = 13.22 ; ;

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{ 17**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 39° 1'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-17**2-27**2 }{ 2 * 17 * 27 } ) = 51° 3'27" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-17**2-21**2 }{ 2 * 21 * 17 } ) = 89° 55'11" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 178.5 }{ 32.5 } = 5.49 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 17 }{ 2 * sin 39° 1'22" } = 13.5 ; ;




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