4 19 21 triangle

Obtuse scalene triangle.

Sides: a = 4   b = 19   c = 21

Area: T = 34.46773758792
Perimeter: p = 44
Semiperimeter: s = 22

Angle ∠ A = α = 9.94988422087° = 9°56'56″ = 0.17436400533 rad
Angle ∠ B = β = 55.1550095421° = 55°9' = 0.96325507479 rad
Angle ∠ C = γ = 114.901106237° = 114°54'4″ = 2.00554018524 rad

Height: ha = 17.23436879396
Height: hb = 3.62881448294
Height: hc = 3.28326072266

Median: ma = 19.92548588452
Median: mb = 11.75879760163
Median: mc = 8.84659030065

Inradius: r = 1.56766989036
Circumradius: R = 11.57661641211

Vertex coordinates: A[21; 0] B[0; 0] C[2.28657142857; 3.28326072266]
Centroid: CG[7.76219047619; 1.09442024089]
Coordinates of the circumscribed circle: U[10.5; -4.87441743668]
Coordinates of the inscribed circle: I[3; 1.56766989036]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170.0511157791° = 170°3'4″ = 0.17436400533 rad
∠ B' = β' = 124.8549904579° = 124°51' = 0.96325507479 rad
∠ C' = γ' = 65.09989376296° = 65°5'56″ = 2.00554018524 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 = 4 ; ; b = 19 ; ; c = 21 ; ;

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

p = a+b+c = 4+19+21 = 44 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 44 }{ 2 } = 22 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22 * (22-4)(22-19)(22-21) } ; ; T = sqrt{ 1188 } = 34.47 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 34.47 }{ 4 } = 17.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 34.47 }{ 19 } = 3.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 34.47 }{ 21 } = 3.28 ; ;

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{ 4**2-19**2-21**2 }{ 2 * 19 * 21 } ) = 9° 56'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-4**2-21**2 }{ 2 * 4 * 21 } ) = 55° 9' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-4**2-19**2 }{ 2 * 19 * 4 } ) = 114° 54'4" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 34.47 }{ 22 } = 1.57 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4 }{ 2 * sin 9° 56'56" } = 11.58 ; ;




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