12 21 24 triangle

Acute scalene triangle.

Sides: a = 12   b = 21   c = 24

Area: T = 125.9879909112
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 29.99547255274° = 29°59'41″ = 0.52435067187 rad
Angle ∠ B = β = 61.02884677763° = 61°1'42″ = 1.06551477001 rad
Angle ∠ C = γ = 88.97768066963° = 88°58'37″ = 1.55329382348 rad

Height: ha = 20.99766515188
Height: hb = 11.99880865821
Height: hc = 10.49883257594

Median: ma = 21.73770651193
Median: mb = 15.80334806293
Median: mc = 12.1866057607

Inradius: r = 4.42203476882
Circumradius: R = 12.0021913723

Vertex coordinates: A[24; 0] B[0; 0] C[5.81325; 10.49883257594]
Centroid: CG[9.93875; 3.49994419198]
Coordinates of the circumscribed circle: U[12; 0.21443198879]
Coordinates of the inscribed circle: I[7.5; 4.42203476882]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0055274473° = 150°19″ = 0.52435067187 rad
∠ B' = β' = 118.9721532224° = 118°58'18″ = 1.06551477001 rad
∠ C' = γ' = 91.02331933037° = 91°1'23″ = 1.55329382348 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 = 12 ; ; b = 21 ; ; c = 24 ; ;

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

p = a+b+c = 12+21+24 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-12)(28.5-21)(28.5-24) } ; ; T = sqrt{ 15870.94 } = 125.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 125.98 }{ 12 } = 21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 125.98 }{ 21 } = 12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 125.98 }{ 24 } = 10.5 ; ;

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{ 12**2-21**2-24**2 }{ 2 * 21 * 24 } ) = 29° 59'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-12**2-24**2 }{ 2 * 12 * 24 } ) = 61° 1'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-12**2-21**2 }{ 2 * 21 * 12 } ) = 88° 58'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 125.98 }{ 28.5 } = 4.42 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 29° 59'41" } = 12 ; ;




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