21 25 29 triangle

Acute scalene triangle.

Sides: a = 21   b = 25   c = 29

Area: T = 256.402239371
Perimeter: p = 75
Semiperimeter: s = 37.5

Angle ∠ A = α = 45.01770320401° = 45°1'1″ = 0.78656954286 rad
Angle ∠ B = β = 57.35765930761° = 57°21'24″ = 1.00110613969 rad
Angle ∠ C = γ = 77.62663748838° = 77°37'35″ = 1.35548358281 rad

Height: ha = 24.41992755914
Height: hb = 20.51221914968
Height: hc = 17.68329237041

Median: ma = 24.95549594269
Median: mb = 22.01770388563
Median: mc = 17.96552442232

Inradius: r = 6.83773971656
Circumradius: R = 14.84548302098

Vertex coordinates: A[29; 0] B[0; 0] C[11.32875862069; 17.68329237041]
Centroid: CG[13.44325287356; 5.89443079014]
Coordinates of the circumscribed circle: U[14.5; 3.1811035045]
Coordinates of the inscribed circle: I[12.5; 6.83773971656]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.983296796° = 134°58'59″ = 0.78656954286 rad
∠ B' = β' = 122.6433406924° = 122°38'36″ = 1.00110613969 rad
∠ C' = γ' = 102.3743625116° = 102°22'25″ = 1.35548358281 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 = 21 ; ; b = 25 ; ; c = 29 ; ;

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

p = a+b+c = 21+25+29 = 75 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 75 }{ 2 } = 37.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37.5 * (37.5-21)(37.5-25)(37.5-29) } ; ; T = sqrt{ 65742.19 } = 256.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 256.4 }{ 21 } = 24.42 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 256.4 }{ 25 } = 20.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 256.4 }{ 29 } = 17.68 ; ;

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{ 21**2-25**2-29**2 }{ 2 * 25 * 29 } ) = 45° 1'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 25**2-21**2-29**2 }{ 2 * 21 * 29 } ) = 57° 21'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-21**2-25**2 }{ 2 * 25 * 21 } ) = 77° 37'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 256.4 }{ 37.5 } = 6.84 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 45° 1'1" } = 14.84 ; ;




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