23 24 25 triangle

Acute scalene triangle.

Sides: a = 23   b = 24   c = 25

Area: T = 248.5487782127
Perimeter: p = 72
Semiperimeter: s = 36

Angle ∠ A = α = 55.94442022574° = 55°56'39″ = 0.97664105268 rad
Angle ∠ B = β = 59.82772594831° = 59°49'38″ = 1.04441826604 rad
Angle ∠ C = γ = 64.22985382594° = 64°13'43″ = 1.12109994664 rad

Height: ha = 21.61328506197
Height: hb = 20.71223151772
Height: hc = 19.88438225701

Median: ma = 21.63990850084
Median: mb = 20.80986520467
Median: mc = 19.90660292374

Inradius: r = 6.90441050591
Circumradius: R = 13.88106308006

Vertex coordinates: A[25; 0] B[0; 0] C[11.56; 19.88438225701]
Centroid: CG[12.18766666667; 6.62879408567]
Coordinates of the circumscribed circle: U[12.5; 6.03550568698]
Coordinates of the inscribed circle: I[12; 6.90441050591]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.0565797743° = 124°3'21″ = 0.97664105268 rad
∠ B' = β' = 120.1732740517° = 120°10'22″ = 1.04441826604 rad
∠ C' = γ' = 115.7711461741° = 115°46'17″ = 1.12109994664 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 = 24 ; ; c = 25 ; ;

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

p = a+b+c = 23+24+25 = 72 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72 }{ 2 } = 36 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36 * (36-23)(36-24)(36-25) } ; ; T = sqrt{ 61776 } = 248.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 248.55 }{ 23 } = 21.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 248.55 }{ 24 } = 20.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 248.55 }{ 25 } = 19.88 ; ;

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**2-24**2-25**2 }{ 2 * 24 * 25 } ) = 55° 56'39" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-23**2-25**2 }{ 2 * 23 * 25 } ) = 59° 49'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-23**2-24**2 }{ 2 * 24 * 23 } ) = 64° 13'43" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 248.55 }{ 36 } = 6.9 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 55° 56'39" } = 13.88 ; ;




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