23 27 28 triangle

Acute scalene triangle.

Sides: a = 23   b = 27   c = 28

Area: T = 286.9988257834
Perimeter: p = 78
Semiperimeter: s = 39

Angle ∠ A = α = 49.39985335° = 49°23'55″ = 0.86221670552 rad
Angle ∠ B = β = 63.03769273045° = 63°2'13″ = 1.11002019318 rad
Angle ∠ C = γ = 67.56545391954° = 67°33'52″ = 1.17992236666 rad

Height: ha = 24.95663702465
Height: hb = 21.259913021
Height: hc = 20.54998755596

Median: ma = 24.98549954973
Median: mb = 21.77772817404
Median: mc = 20.80986520467

Inradius: r = 7.35989296881
Circumradius: R = 15.14664334063

Vertex coordinates: A[28; 0] B[0; 0] C[10.42985714286; 20.54998755596]
Centroid: CG[12.81095238095; 6.83332918532]
Coordinates of the circumscribed circle: U[14; 5.78105228942]
Coordinates of the inscribed circle: I[12; 7.35989296881]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.60114665° = 130°36'5″ = 0.86221670552 rad
∠ B' = β' = 116.9633072695° = 116°57'47″ = 1.11002019318 rad
∠ C' = γ' = 112.4355460805° = 112°26'8″ = 1.17992236666 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 = 23 ; ; b = 27 ; ; c = 28 ; ;

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

p = a+b+c = 23+27+28 = 78 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 78 }{ 2 } = 39 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39 * (39-23)(39-27)(39-28) } ; ; T = sqrt{ 82368 } = 287 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 287 }{ 23 } = 24.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 287 }{ 27 } = 21.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 287 }{ 28 } = 20.5 ; ;

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{ 23**2-27**2-28**2 }{ 2 * 27 * 28 } ) = 49° 23'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-23**2-28**2 }{ 2 * 23 * 28 } ) = 63° 2'13" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-23**2-27**2 }{ 2 * 27 * 23 } ) = 67° 33'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 287 }{ 39 } = 7.36 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 49° 23'55" } = 15.15 ; ;




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