10 21 29 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 21   c = 29

Area: T = 73.48546922835
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 13.96549945758° = 13°57'54″ = 0.24437351354 rad
Angle ∠ B = β = 30.45503140214° = 30°27'1″ = 0.53114582379 rad
Angle ∠ C = γ = 135.5854691403° = 135°35'5″ = 2.36663992803 rad

Height: ha = 14.69769384567
Height: hb = 6.99985421222
Height: hc = 5.06879098127

Median: ma = 24.8199347292
Median: mb = 18.98802528961
Median: mc = 7.76220873481

Inradius: r = 2.44994897428
Circumradius: R = 20.7198600741

Vertex coordinates: A[29; 0] B[0; 0] C[8.62106896552; 5.06879098127]
Centroid: CG[12.54402298851; 1.68993032709]
Coordinates of the circumscribed circle: U[14.5; -14.79990005293]
Coordinates of the inscribed circle: I[9; 2.44994897428]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 166.0355005424° = 166°2'6″ = 0.24437351354 rad
∠ B' = β' = 149.5549685979° = 149°32'59″ = 0.53114582379 rad
∠ C' = γ' = 44.41553085972° = 44°24'55″ = 2.36663992803 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 = 10 ; ; b = 21 ; ; c = 29 ; ;

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

p = a+b+c = 10+21+29 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-10)(30-21)(30-29) } ; ; T = sqrt{ 5400 } = 73.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 73.48 }{ 10 } = 14.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 73.48 }{ 21 } = 7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 73.48 }{ 29 } = 5.07 ; ;

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{ 10**2-21**2-29**2 }{ 2 * 21 * 29 } ) = 13° 57'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-10**2-29**2 }{ 2 * 10 * 29 } ) = 30° 27'1" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-10**2-21**2 }{ 2 * 21 * 10 } ) = 135° 35'5" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 73.48 }{ 30 } = 2.45 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 13° 57'54" } = 20.72 ; ;




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