23 24 24 triangle

Acute isosceles triangle.

Sides: a = 23   b = 24   c = 24

Area: T = 242.2521805979
Perimeter: p = 71
Semiperimeter: s = 35.5

Angle ∠ A = α = 57.26219796738° = 57°15'43″ = 0.99994100815 rad
Angle ∠ B = β = 61.36990101631° = 61°22'8″ = 1.0711091286 rad
Angle ∠ C = γ = 61.36990101631° = 61°22'8″ = 1.0711091286 rad

Height: ha = 21.06553744329
Height: hb = 20.18876504982
Height: hc = 20.18876504982

Median: ma = 21.06553744329
Median: mb = 20.21113829314
Median: mc = 20.21113829314

Inradius: r = 6.82439945346
Circumradius: R = 13.67217247024

Vertex coordinates: A[24; 0] B[0; 0] C[11.02108333333; 20.18876504982]
Centroid: CG[11.67436111111; 6.72992168327]
Coordinates of the circumscribed circle: U[12; 6.55110347532]
Coordinates of the inscribed circle: I[11.5; 6.82439945346]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 122.7388020326° = 122°44'17″ = 0.99994100815 rad
∠ B' = β' = 118.6310989837° = 118°37'52″ = 1.0711091286 rad
∠ C' = γ' = 118.6310989837° = 118°37'52″ = 1.0711091286 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 = 24 ; ;

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

p = a+b+c = 23+24+24 = 71 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 71 }{ 2 } = 35.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35.5 * (35.5-23)(35.5-24)(35.5-24) } ; ; T = sqrt{ 58685.94 } = 242.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 242.25 }{ 23 } = 21.07 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 242.25 }{ 24 } = 20.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 242.25 }{ 24 } = 20.19 ; ;

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**2-24**2 }{ 2 * 24 * 24 } ) = 57° 15'43" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-23**2-24**2 }{ 2 * 23 * 24 } ) = 61° 22'8" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-23**2-24**2 }{ 2 * 24 * 23 } ) = 61° 22'8" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 242.25 }{ 35.5 } = 6.82 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 57° 15'43" } = 13.67 ; ;




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