14 17 20 triangle

Acute scalene triangle.

Sides: a = 14   b = 17   c = 20

Area: T = 117.0877307169
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 43.53111521674° = 43°31'52″ = 0.76597619325 rad
Angle ∠ B = β = 56.75554084441° = 56°45'19″ = 0.99105687457 rad
Angle ∠ C = γ = 79.71334393885° = 79°42'48″ = 1.39112619754 rad

Height: ha = 16.72767581669
Height: hb = 13.7754977314
Height: hc = 11.70987307169

Median: ma = 17.19901134377
Median: mb = 15.02549792013
Median: mc = 11.93773363863

Inradius: r = 4.59216591047
Circumradius: R = 10.1633356121

Vertex coordinates: A[20; 0] B[0; 0] C[7.675; 11.70987307169]
Centroid: CG[9.225; 3.9032910239]
Coordinates of the circumscribed circle: U[10; 1.81548850216]
Coordinates of the inscribed circle: I[8.5; 4.59216591047]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.4698847833° = 136°28'8″ = 0.76597619325 rad
∠ B' = β' = 123.2454591556° = 123°14'41″ = 0.99105687457 rad
∠ C' = γ' = 100.2876560611° = 100°17'12″ = 1.39112619754 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 = 17 ; ; c = 20 ; ;

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

p = a+b+c = 14+17+20 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-14)(25.5-17)(25.5-20) } ; ; T = sqrt{ 13709.44 } = 117.09 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 117.09 }{ 14 } = 16.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 117.09 }{ 17 } = 13.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 117.09 }{ 20 } = 11.71 ; ;

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**2-17**2-20**2 }{ 2 * 17 * 20 } ) = 43° 31'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-14**2-20**2 }{ 2 * 14 * 20 } ) = 56° 45'19" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-14**2-17**2 }{ 2 * 17 * 14 } ) = 79° 42'48" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 117.09 }{ 25.5 } = 4.59 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 43° 31'52" } = 10.16 ; ;




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