21 28 29 triangle

Acute scalene triangle.

Sides: a = 21   b = 28   c = 29

Area: T = 277.8854868246
Perimeter: p = 78
Semiperimeter: s = 39

Angle ∠ A = α = 43.19220135372° = 43°11'31″ = 0.75438428468 rad
Angle ∠ B = β = 65.86663189457° = 65°51'59″ = 1.15495841318 rad
Angle ∠ C = γ = 70.94216675171° = 70°56'30″ = 1.2388165675 rad

Height: ha = 26.46552255472
Height: hb = 19.84989191604
Height: hc = 19.16444736721

Median: ma = 26.5
Median: mb = 21.09550231097
Median: mc = 20.05661711201

Inradius: r = 7.12552530319
Circumradius: R = 15.34108856945

Vertex coordinates: A[29; 0] B[0; 0] C[8.58662068966; 19.16444736721]
Centroid: CG[12.52987356322; 6.38881578907]
Coordinates of the circumscribed circle: U[14.5; 5.00992687982]
Coordinates of the inscribed circle: I[11; 7.12552530319]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.8087986463° = 136°48'29″ = 0.75438428468 rad
∠ B' = β' = 114.1343681054° = 114°8'1″ = 1.15495841318 rad
∠ C' = γ' = 109.0588332483° = 109°3'30″ = 1.2388165675 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 = 28 ; ; c = 29 ; ;

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

p = a+b+c = 21+28+29 = 78 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 78 }{ 2 } = 39 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39 * (39-21)(39-28)(39-29) } ; ; T = sqrt{ 77220 } = 277.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 277.88 }{ 21 } = 26.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 277.88 }{ 28 } = 19.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 277.88 }{ 29 } = 19.16 ; ;

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**2-28**2-29**2 }{ 2 * 28 * 29 } ) = 43° 11'31" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28**2-21**2-29**2 }{ 2 * 21 * 29 } ) = 65° 51'59" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-21**2-28**2 }{ 2 * 28 * 21 } ) = 70° 56'30" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 277.88 }{ 39 } = 7.13 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 43° 11'31" } = 15.34 ; ;




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