5 23 24 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 23   c = 24

Area: T = 57.2366352085
Perimeter: p = 52
Semiperimeter: s = 26

Angle ∠ A = α = 11.96987455547° = 11°58'8″ = 0.20988940173 rad
Angle ∠ B = β = 72.54223968763° = 72°32'33″ = 1.26661036728 rad
Angle ∠ C = γ = 95.4898857569° = 95°29'20″ = 1.66765949635 rad

Height: ha = 22.8954540834
Height: hb = 4.97770740943
Height: hc = 4.77696960071

Median: ma = 23.3721991785
Median: mb = 12.97111217711
Median: mc = 11.53325625947

Inradius: r = 2.20113981571
Circumradius: R = 12.05552756223

Vertex coordinates: A[24; 0] B[0; 0] C[1.5; 4.77696960071]
Centroid: CG[8.5; 1.5989898669]
Coordinates of the circumscribed circle: U[12; -1.15331133204]
Coordinates of the inscribed circle: I[3; 2.20113981571]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 168.0311254445° = 168°1'53″ = 0.20988940173 rad
∠ B' = β' = 107.4587603124° = 107°27'27″ = 1.26661036728 rad
∠ C' = γ' = 84.5111142431° = 84°30'40″ = 1.66765949635 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 = 5 ; ; b = 23 ; ; c = 24 ; ;

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

p = a+b+c = 5+23+24 = 52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 52 }{ 2 } = 26 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26 * (26-5)(26-23)(26-24) } ; ; T = sqrt{ 3276 } = 57.24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 57.24 }{ 5 } = 22.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 57.24 }{ 23 } = 4.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 57.24 }{ 24 } = 4.77 ; ;

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{ 5**2-23**2-24**2 }{ 2 * 23 * 24 } ) = 11° 58'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-5**2-24**2 }{ 2 * 5 * 24 } ) = 72° 32'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-5**2-23**2 }{ 2 * 23 * 5 } ) = 95° 29'20" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 57.24 }{ 26 } = 2.2 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 11° 58'8" } = 12.06 ; ;




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