9 22 24 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 22   c = 24

Area: T = 98.9621798185
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 22.01553695404° = 22°55″ = 0.38442406845 rad
Angle ∠ B = β = 66.3932876283° = 66°23'34″ = 1.1598774291 rad
Angle ∠ C = γ = 91.59217541766° = 91°35'30″ = 1.59985776781 rad

Height: ha = 21.99215107078
Height: hb = 8.99765271077
Height: hc = 8.24768165154

Median: ma = 22.57876438098
Median: mb = 14.40548602909
Median: mc = 11.76986022959

Inradius: r = 3.59986108431
Circumradius: R = 12.00546323105

Vertex coordinates: A[24; 0] B[0; 0] C[3.60441666667; 8.24768165154]
Centroid: CG[9.20113888889; 2.74989388385]
Coordinates of the circumscribed circle: U[12; -0.33334620086]
Coordinates of the inscribed circle: I[5.5; 3.59986108431]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.985463046° = 157°59'5″ = 0.38442406845 rad
∠ B' = β' = 113.6077123717° = 113°36'26″ = 1.1598774291 rad
∠ C' = γ' = 88.40882458234° = 88°24'30″ = 1.59985776781 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 = 9 ; ; b = 22 ; ; c = 24 ; ;

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

p = a+b+c = 9+22+24 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-9)(27.5-22)(27.5-24) } ; ; T = sqrt{ 9793.44 } = 98.96 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 98.96 }{ 9 } = 21.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 98.96 }{ 22 } = 9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 98.96 }{ 24 } = 8.25 ; ;

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{ 9**2-22**2-24**2 }{ 2 * 22 * 24 } ) = 22° 55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-9**2-24**2 }{ 2 * 9 * 24 } ) = 66° 23'34" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-9**2-22**2 }{ 2 * 22 * 9 } ) = 91° 35'30" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 98.96 }{ 27.5 } = 3.6 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 22° 55" } = 12 ; ;




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