14 23 24 triangle

Acute scalene triangle.

Sides: a = 14   b = 23   c = 24

Area: T = 156.6321534181
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 34.57664633043° = 34°34'35″ = 0.60334731284 rad
Angle ∠ B = β = 68.80110709829° = 68°48'4″ = 1.20108052175 rad
Angle ∠ C = γ = 76.62224657128° = 76°37'21″ = 1.33773143077 rad

Height: ha = 22.37659334545
Height: hb = 13.62201334071
Height: hc = 13.05326278484

Median: ma = 22.43988056723
Median: mb = 15.93295323221
Median: mc = 14.78217454991

Inradius: r = 5.13554601371
Circumradius: R = 12.33546809447

Vertex coordinates: A[24; 0] B[0; 0] C[5.06325; 13.05326278484]
Centroid: CG[9.68875; 4.35108759495]
Coordinates of the circumscribed circle: U[12; 2.85438314608]
Coordinates of the inscribed circle: I[7.5; 5.13554601371]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.4243536696° = 145°25'25″ = 0.60334731284 rad
∠ B' = β' = 111.1998929017° = 111°11'56″ = 1.20108052175 rad
∠ C' = γ' = 103.3787534287° = 103°22'39″ = 1.33773143077 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 = 14 ; ; b = 23 ; ; c = 24 ; ;

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

p = a+b+c = 14+23+24 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-14)(30.5-23)(30.5-24) } ; ; T = sqrt{ 24533.44 } = 156.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 156.63 }{ 14 } = 22.38 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 156.63 }{ 23 } = 13.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 156.63 }{ 24 } = 13.05 ; ;

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{ 14**2-23**2-24**2 }{ 2 * 23 * 24 } ) = 34° 34'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-14**2-24**2 }{ 2 * 14 * 24 } ) = 68° 48'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-14**2-23**2 }{ 2 * 23 * 14 } ) = 76° 37'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 156.63 }{ 30.5 } = 5.14 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 34° 34'35" } = 12.33 ; ;




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