13 27 29 triangle

Acute scalene triangle.

Sides: a = 13   b = 27   c = 29

Area: T = 174.9210517665
Perimeter: p = 69
Semiperimeter: s = 34.5

Angle ∠ A = α = 26.53882846718° = 26°32'18″ = 0.46331804454 rad
Angle ∠ B = β = 68.11990638941° = 68°7'9″ = 1.18989019483 rad
Angle ∠ C = γ = 85.34326514341° = 85°20'34″ = 1.49895102599 rad

Height: ha = 26.91108488715
Height: hb = 12.95770753826
Height: hc = 12.06334839769

Median: ma = 27.25334401498
Median: mb = 17.96552442232
Median: mc = 15.45215371404

Inradius: r = 5.07701599323
Circumradius: R = 14.54880360679

Vertex coordinates: A[29; 0] B[0; 0] C[4.84548275862; 12.06334839769]
Centroid: CG[11.28216091954; 4.02111613256]
Coordinates of the circumscribed circle: U[14.5; 1.18112507918]
Coordinates of the inscribed circle: I[7.5; 5.07701599323]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.4621715328° = 153°27'42″ = 0.46331804454 rad
∠ B' = β' = 111.8810936106° = 111°52'51″ = 1.18989019483 rad
∠ C' = γ' = 94.65773485659° = 94°39'26″ = 1.49895102599 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 = 27 ; ; c = 29 ; ;

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

p = a+b+c = 13+27+29 = 69 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 69 }{ 2 } = 34.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.5 * (34.5-13)(34.5-27)(34.5-29) } ; ; T = sqrt{ 30597.19 } = 174.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 174.92 }{ 13 } = 26.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 174.92 }{ 27 } = 12.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 174.92 }{ 29 } = 12.06 ; ;

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**2-27**2-29**2 }{ 2 * 27 * 29 } ) = 26° 32'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-13**2-29**2 }{ 2 * 13 * 29 } ) = 68° 7'9" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-13**2-27**2 }{ 2 * 27 * 13 } ) = 85° 20'34" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 174.92 }{ 34.5 } = 5.07 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 26° 32'18" } = 14.55 ; ;




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