15 27 29 triangle

Acute scalene triangle.

Sides: a = 15   b = 27   c = 29

Area: T = 200.5219793287
Perimeter: p = 71
Semiperimeter: s = 35.5

Angle ∠ A = α = 30.80993733352° = 30°48'34″ = 0.53877250052 rad
Angle ∠ B = β = 67.21098982792° = 67°12'36″ = 1.17330340149 rad
Angle ∠ C = γ = 81.98107283856° = 81°58'51″ = 1.43108336335 rad

Height: ha = 26.73659724383
Height: hb = 14.85333180213
Height: hc = 13.82989512612

Median: ma = 26.99553699734
Median: mb = 18.72883208003
Median: mc = 16.33224829711

Inradius: r = 5.64884448813
Circumradius: R = 14.64331928333

Vertex coordinates: A[29; 0] B[0; 0] C[5.81103448276; 13.82989512612]
Centroid: CG[11.60334482759; 4.61096504204]
Coordinates of the circumscribed circle: U[14.5; 2.04328157903]
Coordinates of the inscribed circle: I[8.5; 5.64884448813]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.1910626665° = 149°11'26″ = 0.53877250052 rad
∠ B' = β' = 112.7990101721° = 112°47'24″ = 1.17330340149 rad
∠ C' = γ' = 98.01992716144° = 98°1'9″ = 1.43108336335 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 = 15 ; ; b = 27 ; ; c = 29 ; ;

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

p = a+b+c = 15+27+29 = 71 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 71 }{ 2 } = 35.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35.5 * (35.5-15)(35.5-27)(35.5-29) } ; ; T = sqrt{ 40208.19 } = 200.52 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 200.52 }{ 15 } = 26.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 200.52 }{ 27 } = 14.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 200.52 }{ 29 } = 13.83 ; ;

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{ 15**2-27**2-29**2 }{ 2 * 27 * 29 } ) = 30° 48'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-15**2-29**2 }{ 2 * 15 * 29 } ) = 67° 12'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-15**2-27**2 }{ 2 * 27 * 15 } ) = 81° 58'51" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 200.52 }{ 35.5 } = 5.65 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 30° 48'34" } = 14.64 ; ;




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