21 22 29 triangle

Acute scalene triangle.

Sides: a = 21   b = 22   c = 29

Area: T = 230.0433474152
Perimeter: p = 72
Semiperimeter: s = 36

Angle ∠ A = α = 46.14986331446° = 46°8'55″ = 0.80554455937 rad
Angle ∠ B = β = 49.06772754258° = 49°4'2″ = 0.85663855112 rad
Angle ∠ C = γ = 84.78440914295° = 84°47'3″ = 1.48797615488 rad

Height: ha = 21.90989023002
Height: hb = 20.91330431047
Height: hc = 15.86550671829

Median: ma = 23.5
Median: mb = 22.8043508502
Median: mc = 15.88223801743

Inradius: r = 6.39900965042
Circumradius: R = 14.56602913203

Vertex coordinates: A[29; 0] B[0; 0] C[13.75986206897; 15.86550671829]
Centroid: CG[14.25328735632; 5.28883557276]
Coordinates of the circumscribed circle: U[14.5; 1.32436628473]
Coordinates of the inscribed circle: I[14; 6.39900965042]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.8511366855° = 133°51'5″ = 0.80554455937 rad
∠ B' = β' = 130.9332724574° = 130°55'58″ = 0.85663855112 rad
∠ C' = γ' = 95.21659085705° = 95°12'57″ = 1.48797615488 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 = 22 ; ; c = 29 ; ;

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

p = a+b+c = 21+22+29 = 72 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72 }{ 2 } = 36 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36 * (36-21)(36-22)(36-29) } ; ; T = sqrt{ 52920 } = 230.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 230.04 }{ 21 } = 21.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 230.04 }{ 22 } = 20.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 230.04 }{ 29 } = 15.87 ; ;

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**2-22**2-29**2 }{ 2 * 22 * 29 } ) = 46° 8'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-21**2-29**2 }{ 2 * 21 * 29 } ) = 49° 4'2" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-21**2-22**2 }{ 2 * 22 * 21 } ) = 84° 47'3" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 230.04 }{ 36 } = 6.39 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 46° 8'55" } = 14.56 ; ;




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