10 21 24 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 21   c = 24

Area: T = 104.6354781502
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 24.5333007117° = 24°31'59″ = 0.42881817496 rad
Angle ∠ B = β = 60.68768010358° = 60°41'12″ = 1.05991844906 rad
Angle ∠ C = γ = 94.78801918472° = 94°46'49″ = 1.65442264134 rad

Height: ha = 20.92769563004
Height: hb = 9.96552172859
Height: hc = 8.72195651252

Median: ma = 21.98986334273
Median: mb = 15.09113882728
Median: mc = 11.24772218792

Inradius: r = 3.80549011455
Circumradius: R = 12.04218849441

Vertex coordinates: A[24; 0] B[0; 0] C[4.89658333333; 8.72195651252]
Centroid: CG[9.63219444444; 2.90765217084]
Coordinates of the circumscribed circle: U[12; -1.0033490412]
Coordinates of the inscribed circle: I[6.5; 3.80549011455]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.4676992883° = 155°28'1″ = 0.42881817496 rad
∠ B' = β' = 119.3133198964° = 119°18'48″ = 1.05991844906 rad
∠ C' = γ' = 85.22198081528° = 85°13'11″ = 1.65442264134 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 = 21 ; ; c = 24 ; ;

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

p = a+b+c = 10+21+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 * (27.5-10)(27.5-21)(27.5-24) } ; ; T = sqrt{ 10948.44 } = 104.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 104.63 }{ 10 } = 20.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 104.63 }{ 21 } = 9.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 104.63 }{ 24 } = 8.72 ; ;

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**2-21**2-24**2 }{ 2 * 21 * 24 } ) = 24° 31'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-10**2-24**2 }{ 2 * 10 * 24 } ) = 60° 41'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-10**2-21**2 }{ 2 * 21 * 10 } ) = 94° 46'49" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 104.63 }{ 27.5 } = 3.8 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 24° 31'59" } = 12.04 ; ;




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