27 28 30 triangle

Acute scalene triangle.

Sides: a = 27   b = 28   c = 30

Area: T = 345.5410789922
Perimeter: p = 85
Semiperimeter: s = 42.5

Angle ∠ A = α = 55.3587624069° = 55°21'27″ = 0.96661728061 rad
Angle ∠ B = β = 58.56600307183° = 58°33'36″ = 1.02220653461 rad
Angle ∠ C = γ = 66.08223452127° = 66°4'56″ = 1.15333545014 rad

Height: ha = 25.59656140683
Height: hb = 24.68114849944
Height: hc = 23.03660526615

Median: ma = 25.68655990781
Median: mb = 24.87696602309
Median: mc = 23.05442837668

Inradius: r = 8.13303715276
Circumradius: R = 16.40990612899

Vertex coordinates: A[30; 0] B[0; 0] C[14.08333333333; 23.03660526615]
Centroid: CG[14.69444444444; 7.67986842205]
Coordinates of the circumscribed circle: U[15; 6.65326154568]
Coordinates of the inscribed circle: I[14.5; 8.13303715276]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.6422375931° = 124°38'33″ = 0.96661728061 rad
∠ B' = β' = 121.4439969282° = 121°26'24″ = 1.02220653461 rad
∠ C' = γ' = 113.9187654787° = 113°55'4″ = 1.15333545014 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 = 28 ; ; c = 30 ; ;

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

p = a+b+c = 27+28+30 = 85 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 85 }{ 2 } = 42.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 42.5 * (42.5-27)(42.5-28)(42.5-30) } ; ; T = sqrt{ 119398.44 } = 345.54 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 345.54 }{ 27 } = 25.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 345.54 }{ 28 } = 24.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 345.54 }{ 30 } = 23.04 ; ;

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**2-28**2-30**2 }{ 2 * 28 * 30 } ) = 55° 21'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28**2-27**2-30**2 }{ 2 * 27 * 30 } ) = 58° 33'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-27**2-28**2 }{ 2 * 28 * 27 } ) = 66° 4'56" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 345.54 }{ 42.5 } = 8.13 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 27 }{ 2 * sin 55° 21'27" } = 16.41 ; ;




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