19 27 28 triangle

Acute scalene triangle.

Sides: a = 19   b = 27   c = 28

Area: T = 244.8266469157
Perimeter: p = 74
Semiperimeter: s = 37

Angle ∠ A = α = 40.36875935442° = 40°22'3″ = 0.70545474185 rad
Angle ∠ B = β = 66.9854667524° = 66°59'5″ = 1.16991029966 rad
Angle ∠ C = γ = 72.64877389318° = 72°38'52″ = 1.26879422385 rad

Height: ha = 25.77112072797
Height: hb = 18.13552940116
Height: hc = 17.48876049398

Median: ma = 25.81218189983
Median: mb = 19.75547462651
Median: mc = 18.68215416923

Inradius: r = 6.61769315988
Circumradius: R = 14.66875317108

Vertex coordinates: A[28; 0] B[0; 0] C[7.42985714286; 17.48876049398]
Centroid: CG[11.81095238095; 5.82992016466]
Coordinates of the circumscribed circle: U[14; 4.37545270015]
Coordinates of the inscribed circle: I[10; 6.61769315988]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.6322406456° = 139°37'57″ = 0.70545474185 rad
∠ B' = β' = 113.0155332476° = 113°55″ = 1.16991029966 rad
∠ C' = γ' = 107.3522261068° = 107°21'8″ = 1.26879422385 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 = 19 ; ; b = 27 ; ; c = 28 ; ;

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

p = a+b+c = 19+27+28 = 74 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 74 }{ 2 } = 37 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37 * (37-19)(37-27)(37-28) } ; ; T = sqrt{ 59940 } = 244.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 244.83 }{ 19 } = 25.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 244.83 }{ 27 } = 18.14 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 244.83 }{ 28 } = 17.49 ; ;

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{ 19**2-27**2-28**2 }{ 2 * 27 * 28 } ) = 40° 22'3" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-19**2-28**2 }{ 2 * 19 * 28 } ) = 66° 59'5" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-19**2-27**2 }{ 2 * 27 * 19 } ) = 72° 38'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 244.83 }{ 37 } = 6.62 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 40° 22'3" } = 14.67 ; ;




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