19 21 25 triangle

Acute scalene triangle.

Sides: a = 19   b = 21   c = 25

Area: T = 194.531068524
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 47.82325813991° = 47°49'21″ = 0.83546615022 rad
Angle ∠ B = β = 54.99224613631° = 54°59'33″ = 0.96597995146 rad
Angle ∠ C = γ = 77.18549572378° = 77°11'6″ = 1.34771316368 rad

Height: ha = 20.47769142358
Height: hb = 18.52767319276
Height: hc = 15.56224548192

Median: ma = 21.04216254125
Median: mb = 19.56439975465
Median: mc = 15.64444878472

Inradius: r = 5.98655595459
Circumradius: R = 12.81993143253

Vertex coordinates: A[25; 0] B[0; 0] C[10.9; 15.56224548192]
Centroid: CG[11.96766666667; 5.18774849397]
Coordinates of the circumscribed circle: U[12.5; 2.84333817488]
Coordinates of the inscribed circle: I[11.5; 5.98655595459]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.1777418601° = 132°10'39″ = 0.83546615022 rad
∠ B' = β' = 125.0087538637° = 125°27″ = 0.96597995146 rad
∠ C' = γ' = 102.8155042762° = 102°48'54″ = 1.34771316368 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 = 19 ; ; b = 21 ; ; c = 25 ; ;

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

p = a+b+c = 19+21+25 = 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 * (32.5-19)(32.5-21)(32.5-25) } ; ; T = sqrt{ 37842.19 } = 194.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 194.53 }{ 19 } = 20.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 194.53 }{ 21 } = 18.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 194.53 }{ 25 } = 15.56 ; ;

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{ 19**2-21**2-25**2 }{ 2 * 21 * 25 } ) = 47° 49'21" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-19**2-25**2 }{ 2 * 19 * 25 } ) = 54° 59'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-19**2-21**2 }{ 2 * 21 * 19 } ) = 77° 11'6" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 194.53 }{ 32.5 } = 5.99 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 47° 49'21" } = 12.82 ; ;




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