17 20 21 triangle

Acute scalene triangle.

Sides: a = 17   b = 20   c = 21

Area: T = 158.2910871499
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 48.91876668595° = 48°55'4″ = 0.85437743491 rad
Angle ∠ B = β = 62.47218164666° = 62°28'19″ = 1.0990338887 rad
Angle ∠ C = γ = 68.61105166738° = 68°36'38″ = 1.19774794175 rad

Height: ha = 18.62224554705
Height: hb = 15.82990871499
Height: hc = 15.07553210952

Median: ma = 18.66114576065
Median: mb = 16.27988205961
Median: mc = 15.3055227865

Inradius: r = 5.45883059138
Circumradius: R = 11.27767083982

Vertex coordinates: A[21; 0] B[0; 0] C[7.85771428571; 15.07553210952]
Centroid: CG[9.6199047619; 5.02551070317]
Coordinates of the circumscribed circle: U[10.5; 4.11326818864]
Coordinates of the inscribed circle: I[9; 5.45883059138]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.082233314° = 131°4'56″ = 0.85437743491 rad
∠ B' = β' = 117.5288183533° = 117°31'41″ = 1.0990338887 rad
∠ C' = γ' = 111.3899483326° = 111°23'22″ = 1.19774794175 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 = 20 ; ; c = 21 ; ;

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

p = a+b+c = 17+20+21 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-17)(29-20)(29-21) } ; ; T = sqrt{ 25056 } = 158.29 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 158.29 }{ 17 } = 18.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 158.29 }{ 20 } = 15.83 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 158.29 }{ 21 } = 15.08 ; ;

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**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 48° 55'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-17**2-21**2 }{ 2 * 17 * 21 } ) = 62° 28'19" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-17**2-20**2 }{ 2 * 20 * 17 } ) = 68° 36'38" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 158.29 }{ 29 } = 5.46 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 17 }{ 2 * sin 48° 55'4" } = 11.28 ; ;




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