13 28 29 triangle

Acute scalene triangle.

Sides: a = 13   b = 28   c = 29

Area: T = 179.8333256101
Perimeter: p = 70
Semiperimeter: s = 35

Angle ∠ A = α = 26.29215552747° = 26°17'30″ = 0.4598874205 rad
Angle ∠ B = β = 72.55883278237° = 72°33'30″ = 1.26663817203 rad
Angle ∠ C = γ = 81.15501169016° = 81°9' = 1.41663367283 rad

Height: ha = 27.66766547848
Height: hb = 12.84552325787
Height: hc = 12.40222935242

Median: ma = 27.75333781728
Median: mb = 17.57883958312
Median: mc = 16.31771688721

Inradius: r = 5.13880930315
Circumradius: R = 14.67547050974

Vertex coordinates: A[29; 0] B[0; 0] C[3.89765517241; 12.40222935242]
Centroid: CG[10.96655172414; 4.13440978414]
Coordinates of the circumscribed circle: U[14.5; 2.25876469381]
Coordinates of the inscribed circle: I[7; 5.13880930315]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.7088444725° = 153°42'30″ = 0.4598874205 rad
∠ B' = β' = 107.4421672176° = 107°26'30″ = 1.26663817203 rad
∠ C' = γ' = 98.85498830984° = 98°51' = 1.41663367283 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 = 13 ; ; b = 28 ; ; c = 29 ; ;

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

p = a+b+c = 13+28+29 = 70 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 70 }{ 2 } = 35 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35 * (35-13)(35-28)(35-29) } ; ; T = sqrt{ 32340 } = 179.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 179.83 }{ 13 } = 27.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 179.83 }{ 28 } = 12.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 179.83 }{ 29 } = 12.4 ; ;

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{ 13**2-28**2-29**2 }{ 2 * 28 * 29 } ) = 26° 17'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28**2-13**2-29**2 }{ 2 * 13 * 29 } ) = 72° 33'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-13**2-28**2 }{ 2 * 28 * 13 } ) = 81° 9' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 179.83 }{ 35 } = 5.14 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 26° 17'30" } = 14.67 ; ;




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