15 22 24 triangle

Acute scalene triangle.

Sides: a = 15   b = 22   c = 24

Area: T = 161.615509057
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 37.74771807582° = 37°44'50″ = 0.65988125876 rad
Angle ∠ B = β = 63.87883942075° = 63°52'42″ = 1.11548882998 rad
Angle ∠ C = γ = 78.37444250343° = 78°22'28″ = 1.36878917662 rad

Height: ha = 21.54986787427
Height: hb = 14.69222809609
Height: hc = 13.46879242142

Median: ma = 21.76657988597
Median: mb = 16.71882534973
Median: mc = 14.50986181285

Inradius: r = 5.29988554285
Circumradius: R = 12.25113311908

Vertex coordinates: A[24; 0] B[0; 0] C[6.60441666667; 13.46879242142]
Centroid: CG[10.20113888889; 4.48993080714]
Coordinates of the circumscribed circle: U[12; 2.46988288612]
Coordinates of the inscribed circle: I[8.5; 5.29988554285]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.2532819242° = 142°15'10″ = 0.65988125876 rad
∠ B' = β' = 116.1221605793° = 116°7'18″ = 1.11548882998 rad
∠ C' = γ' = 101.6265574966° = 101°37'32″ = 1.36878917662 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 = 15 ; ; b = 22 ; ; c = 24 ; ;

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

p = a+b+c = 15+22+24 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-15)(30.5-22)(30.5-24) } ; ; T = sqrt{ 26119.44 } = 161.62 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 161.62 }{ 15 } = 21.55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 161.62 }{ 22 } = 14.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 161.62 }{ 24 } = 13.47 ; ;

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{ 15**2-22**2-24**2 }{ 2 * 22 * 24 } ) = 37° 44'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-15**2-24**2 }{ 2 * 15 * 24 } ) = 63° 52'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-15**2-22**2 }{ 2 * 22 * 15 } ) = 78° 22'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 161.62 }{ 30.5 } = 5.3 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 37° 44'50" } = 12.25 ; ;




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