8 20 24 triangle

Obtuse scalene triangle.

Sides: a = 8   b = 20   c = 24

Area: T = 74.94399759808
Perimeter: p = 52
Semiperimeter: s = 26

Angle ∠ A = α = 18.19548723388° = 18°11'42″ = 0.31875604293 rad
Angle ∠ B = β = 51.31878125465° = 51°19'4″ = 0.89656647939 rad
Angle ∠ C = γ = 110.4877315115° = 110°29'14″ = 1.92883674304 rad

Height: ha = 18.73549939952
Height: hb = 7.49439975981
Height: hc = 6.24549979984

Median: ma = 21.72655609824
Median: mb = 14.83223969742
Median: mc = 9.38108315196

Inradius: r = 2.88223067685
Circumradius: R = 12.81102523044

Vertex coordinates: A[24; 0] B[0; 0] C[5; 6.24549979984]
Centroid: CG[9.66766666667; 2.08216659995]
Coordinates of the circumscribed circle: U[12; -4.48435883065]
Coordinates of the inscribed circle: I[6; 2.88223067685]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.8055127661° = 161°48'18″ = 0.31875604293 rad
∠ B' = β' = 128.6822187453° = 128°40'56″ = 0.89656647939 rad
∠ C' = γ' = 69.51326848853° = 69°30'46″ = 1.92883674304 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 = 8 ; ; b = 20 ; ; c = 24 ; ;

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

p = a+b+c = 8+20+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 * (26-8)(26-20)(26-24) } ; ; T = sqrt{ 5616 } = 74.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 74.94 }{ 8 } = 18.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 74.94 }{ 20 } = 7.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 74.94 }{ 24 } = 6.24 ; ;

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{ 8**2-20**2-24**2 }{ 2 * 20 * 24 } ) = 18° 11'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-8**2-24**2 }{ 2 * 8 * 24 } ) = 51° 19'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-8**2-20**2 }{ 2 * 20 * 8 } ) = 110° 29'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 74.94 }{ 26 } = 2.88 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 18° 11'42" } = 12.81 ; ;




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