9 23 27 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 23   c = 27

Area: T = 99.13221718717
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 18.61985568494° = 18°37'7″ = 0.3254955119 rad
Angle ∠ B = β = 54.67767118042° = 54°40'36″ = 0.95442886451 rad
Angle ∠ C = γ = 106.7054731346° = 106°42'17″ = 1.86223488895 rad

Height: ha = 22.0299371527
Height: hb = 8.62201888584
Height: hc = 7.34331238423

Median: ma = 24.6732859583
Median: mb = 16.51551445649
Median: mc = 11.07992599031

Inradius: r = 3.36604126058
Circumradius: R = 14.09548188022

Vertex coordinates: A[27; 0] B[0; 0] C[5.20437037037; 7.34331238423]
Centroid: CG[10.73545679012; 2.44877079474]
Coordinates of the circumscribed circle: U[13.5; -4.05114092692]
Coordinates of the inscribed circle: I[6.5; 3.36604126058]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.3811443151° = 161°22'53″ = 0.3254955119 rad
∠ B' = β' = 125.3233288196° = 125°19'24″ = 0.95442886451 rad
∠ C' = γ' = 73.29552686536° = 73°17'43″ = 1.86223488895 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 = 9 ; ; b = 23 ; ; c = 27 ; ;

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

p = a+b+c = 9+23+27 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-9)(29.5-23)(29.5-27) } ; ; T = sqrt{ 9827.19 } = 99.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 99.13 }{ 9 } = 22.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 99.13 }{ 23 } = 8.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 99.13 }{ 27 } = 7.34 ; ;

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{ 9**2-23**2-27**2 }{ 2 * 23 * 27 } ) = 18° 37'7" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-9**2-27**2 }{ 2 * 9 * 27 } ) = 54° 40'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-9**2-23**2 }{ 2 * 23 * 9 } ) = 106° 42'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 99.13 }{ 29.5 } = 3.36 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 18° 37'7" } = 14.09 ; ;




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