10 22 24 triangle

Acute scalene triangle.

Sides: a = 10   b = 22   c = 24

Area: T = 109.9821816679
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 24.62199773287° = 24°37'12″ = 0.43296996662 rad
Angle ∠ B = β = 66.42218215218° = 66°25'19″ = 1.15992794807 rad
Angle ∠ C = γ = 88.95882011495° = 88°57'30″ = 1.55326135067 rad

Height: ha = 21.99663633358
Height: hb = 9.99883469708
Height: hc = 9.16551513899

Median: ma = 22.47222050542
Median: mb = 14.73109198627
Median: mc = 12.16655250606

Inradius: r = 3.92879220242
Circumradius: R = 12.0021983963

Vertex coordinates: A[24; 0] B[0; 0] C[4; 9.16551513899]
Centroid: CG[9.33333333333; 3.05550504633]
Coordinates of the circumscribed circle: U[12; 0.21882178902]
Coordinates of the inscribed circle: I[6; 3.92879220242]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.3880022671° = 155°22'48″ = 0.43296996662 rad
∠ B' = β' = 113.5788178478° = 113°34'41″ = 1.15992794807 rad
∠ C' = γ' = 91.04217988505° = 91°2'30″ = 1.55326135067 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 = 10 ; ; b = 22 ; ; c = 24 ; ;

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

p = a+b+c = 10+22+24 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-10)(28-22)(28-24) } ; ; T = sqrt{ 12096 } = 109.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 109.98 }{ 10 } = 22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 109.98 }{ 22 } = 10 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 109.98 }{ 24 } = 9.17 ; ;

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{ 10**2-22**2-24**2 }{ 2 * 22 * 24 } ) = 24° 37'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-10**2-24**2 }{ 2 * 10 * 24 } ) = 66° 25'19" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-10**2-22**2 }{ 2 * 22 * 10 } ) = 88° 57'30" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 109.98 }{ 28 } = 3.93 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 24° 37'12" } = 12 ; ;




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