23 23 24 triangle

Acute isosceles triangle.

Sides: a = 23   b = 23   c = 24

Area: T = 235.4577002444
Perimeter: p = 70
Semiperimeter: s = 35

Angle ∠ A = α = 58.55110186106° = 58°33'4″ = 1.02219080552 rad
Angle ∠ B = β = 58.55110186106° = 58°33'4″ = 1.02219080552 rad
Angle ∠ C = γ = 62.89879627788° = 62°53'53″ = 1.09877765433 rad

Height: ha = 20.47545219517
Height: hb = 20.47545219517
Height: hc = 19.62114168703

Median: ma = 20.5
Median: mb = 20.5
Median: mc = 19.62114168703

Inradius: r = 6.7277342927
Circumradius: R = 13.48801682135

Vertex coordinates: A[24; 0] B[0; 0] C[12; 19.62114168703]
Centroid: CG[12; 6.54404722901]
Coordinates of the circumscribed circle: U[12; 6.14112486568]
Coordinates of the inscribed circle: I[12; 6.7277342927]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.4498981389° = 121°26'56″ = 1.02219080552 rad
∠ B' = β' = 121.4498981389° = 121°26'56″ = 1.02219080552 rad
∠ C' = γ' = 117.1022037221° = 117°6'7″ = 1.09877765433 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 = 23 ; ; c = 24 ; ;

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

p = a+b+c = 23+23+24 = 70 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 70 }{ 2 } = 35 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35 * (35-23)(35-23)(35-24) } ; ; T = sqrt{ 55440 } = 235.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 235.46 }{ 23 } = 20.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 235.46 }{ 23 } = 20.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 235.46 }{ 24 } = 19.62 ; ;

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**2-23**2-24**2 }{ 2 * 23 * 24 } ) = 58° 33'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-23**2-24**2 }{ 2 * 23 * 24 } ) = 58° 33'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-23**2-23**2 }{ 2 * 23 * 23 } ) = 62° 53'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 235.46 }{ 35 } = 6.73 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 58° 33'4" } = 13.48 ; ;




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