27 29 30 triangle

Acute scalene triangle.

Sides: a = 27   b = 29   c = 30

Area: T = 353.8598728874
Perimeter: p = 86
Semiperimeter: s = 43

Angle ∠ A = α = 54.43661953563° = 54°26'10″ = 0.95500908412 rad
Angle ∠ B = β = 60.8944467291° = 60°53'40″ = 1.06328089505 rad
Angle ∠ C = γ = 64.66993373527° = 64°40'10″ = 1.12986928619 rad

Height: ha = 26.21217576943
Height: hb = 24.40440502671
Height: hc = 23.59105819249

Median: ma = 26.23545192447
Median: mb = 24.58114971066
Median: mc = 23.66443191324

Inradius: r = 8.22992727645
Circumradius: R = 16.59656058755

Vertex coordinates: A[30; 0] B[0; 0] C[13.13333333333; 23.59105819249]
Centroid: CG[14.37877777778; 7.86435273083]
Coordinates of the circumscribed circle: U[15; 7.11002911473]
Coordinates of the inscribed circle: I[14; 8.22992727645]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.5643804644° = 125°33'50″ = 0.95500908412 rad
∠ B' = β' = 119.1065532709° = 119°6'20″ = 1.06328089505 rad
∠ C' = γ' = 115.3310662647° = 115°19'50″ = 1.12986928619 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 = 27 ; ; b = 29 ; ; c = 30 ; ;

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

p = a+b+c = 27+29+30 = 86 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 86 }{ 2 } = 43 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 43 * (43-27)(43-29)(43-30) } ; ; T = sqrt{ 125216 } = 353.86 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 353.86 }{ 27 } = 26.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 353.86 }{ 29 } = 24.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 353.86 }{ 30 } = 23.59 ; ;

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{ 27**2-29**2-30**2 }{ 2 * 29 * 30 } ) = 54° 26'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 29**2-27**2-30**2 }{ 2 * 27 * 30 } ) = 60° 53'40" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-27**2-29**2 }{ 2 * 29 * 27 } ) = 64° 40'10" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 353.86 }{ 43 } = 8.23 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 27 }{ 2 * sin 54° 26'10" } = 16.6 ; ;




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