6 19 21 triangle

Obtuse scalene triangle.

Sides: a = 6   b = 19   c = 21

Area: T = 55.92985258164
Perimeter: p = 46
Semiperimeter: s = 23

Angle ∠ A = α = 16.28107061966° = 16°16'51″ = 0.28441519277 rad
Angle ∠ B = β = 62.59224054672° = 62°35'33″ = 1.09224435621 rad
Angle ∠ C = γ = 101.1276888336° = 101°7'37″ = 1.76549971638 rad

Height: ha = 18.64328419388
Height: hb = 5.88772132438
Height: hc = 5.32765262682

Median: ma = 19.79989898732
Median: mb = 12.17657956619
Median: mc = 9.3944147114

Inradius: r = 2.43216750355
Circumradius: R = 10.70111581525

Vertex coordinates: A[21; 0] B[0; 0] C[2.76219047619; 5.32765262682]
Centroid: CG[7.92106349206; 1.77655087561]
Coordinates of the circumscribed circle: U[10.5; -2.06551357838]
Coordinates of the inscribed circle: I[4; 2.43216750355]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163.7199293803° = 163°43'9″ = 0.28441519277 rad
∠ B' = β' = 117.4087594533° = 117°24'27″ = 1.09224435621 rad
∠ C' = γ' = 78.87331116638° = 78°52'23″ = 1.76549971638 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 = 6 ; ; b = 19 ; ; c = 21 ; ;

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

p = a+b+c = 6+19+21 = 46 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46 }{ 2 } = 23 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23 * (23-6)(23-19)(23-21) } ; ; T = sqrt{ 3128 } = 55.93 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 55.93 }{ 6 } = 18.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 55.93 }{ 19 } = 5.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 55.93 }{ 21 } = 5.33 ; ;

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{ 6**2-19**2-21**2 }{ 2 * 19 * 21 } ) = 16° 16'51" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-6**2-21**2 }{ 2 * 6 * 21 } ) = 62° 35'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-6**2-19**2 }{ 2 * 19 * 6 } ) = 101° 7'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 55.93 }{ 23 } = 2.43 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6 }{ 2 * sin 16° 16'51" } = 10.7 ; ;




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