24 27 29 triangle

Acute scalene triangle.

Sides: a = 24   b = 27   c = 29

Area: T = 302.5232726419
Perimeter: p = 80
Semiperimeter: s = 40

Angle ∠ A = α = 50.59994310494° = 50°35'58″ = 0.88331266714 rad
Angle ∠ B = β = 60.38795039651° = 60°22'46″ = 1.05438211449 rad
Angle ∠ C = γ = 69.02110649855° = 69°1'16″ = 1.20546448372 rad

Height: ha = 25.21102272016
Height: hb = 22.40990908459
Height: hc = 20.86436363048

Median: ma = 25.31879778023
Median: mb = 22.94401394939
Median: mc = 21.03297408448

Inradius: r = 7.56330681605
Circumradius: R = 15.52994118085

Vertex coordinates: A[29; 0] B[0; 0] C[11.86220689655; 20.86436363048]
Centroid: CG[13.62106896552; 6.95545454349]
Coordinates of the circumscribed circle: U[14.5; 5.56599128697]
Coordinates of the inscribed circle: I[13; 7.56330681605]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.4010568951° = 129°24'2″ = 0.88331266714 rad
∠ B' = β' = 119.6220496035° = 119°37'14″ = 1.05438211449 rad
∠ C' = γ' = 110.9798935015° = 110°58'44″ = 1.20546448372 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 = 24 ; ; b = 27 ; ; c = 29 ; ;

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

p = a+b+c = 24+27+29 = 80 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 80 }{ 2 } = 40 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 40 * (40-24)(40-27)(40-29) } ; ; T = sqrt{ 91520 } = 302.52 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 302.52 }{ 24 } = 25.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 302.52 }{ 27 } = 22.41 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 302.52 }{ 29 } = 20.86 ; ;

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{ 24**2-27**2-29**2 }{ 2 * 27 * 29 } ) = 50° 35'58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-24**2-29**2 }{ 2 * 24 * 29 } ) = 60° 22'46" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-24**2-27**2 }{ 2 * 27 * 24 } ) = 69° 1'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 302.52 }{ 40 } = 7.56 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 24 }{ 2 * sin 50° 35'58" } = 15.53 ; ;




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