23 25 27 triangle

Acute scalene triangle.

Sides: a = 23   b = 25   c = 27

Area: T = 267.1466378415
Perimeter: p = 75
Semiperimeter: s = 37.5

Angle ∠ A = α = 52.33301130357° = 52°19'48″ = 0.91333327704 rad
Angle ∠ B = β = 59.35986925825° = 59°21'31″ = 1.03660046252 rad
Angle ∠ C = γ = 68.31111943819° = 68°18'40″ = 1.19222552579 rad

Height: ha = 23.23301198621
Height: hb = 21.37217102732
Height: hc = 19.78986206233

Median: ma = 23.3439880034
Median: mb = 21.74328149052
Median: mc = 19.86883164863

Inradius: r = 7.12439034244
Circumradius: R = 14.52985518113

Vertex coordinates: A[27; 0] B[0; 0] C[11.72222222222; 19.78986206233]
Centroid: CG[12.90774074074; 6.59662068744]
Coordinates of the circumscribed circle: U[13.5; 5.36992474085]
Coordinates of the inscribed circle: I[12.5; 7.12439034244]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.6769886964° = 127°40'12″ = 0.91333327704 rad
∠ B' = β' = 120.6411307418° = 120°38'29″ = 1.03660046252 rad
∠ C' = γ' = 111.6898805618° = 111°41'20″ = 1.19222552579 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 = 25 ; ; c = 27 ; ;

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

p = a+b+c = 23+25+27 = 75 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 75 }{ 2 } = 37.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37.5 * (37.5-23)(37.5-25)(37.5-27) } ; ; T = sqrt{ 71367.19 } = 267.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 267.15 }{ 23 } = 23.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 267.15 }{ 25 } = 21.37 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 267.15 }{ 27 } = 19.79 ; ;

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**2-25**2-27**2 }{ 2 * 25 * 27 } ) = 52° 19'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 25**2-23**2-27**2 }{ 2 * 23 * 27 } ) = 59° 21'31" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-23**2-25**2 }{ 2 * 25 * 23 } ) = 68° 18'40" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 267.15 }{ 37.5 } = 7.12 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 52° 19'48" } = 14.53 ; ;




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