16 21 24 triangle

Acute scalene triangle.

Sides: a = 16   b = 21   c = 24

Area: T = 165.2544160311
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 40.97880974811° = 40°58'41″ = 0.71552027222 rad
Angle ∠ B = β = 59.39551294361° = 59°23'42″ = 1.03766405683 rad
Angle ∠ C = γ = 79.62767730829° = 79°37'36″ = 1.3989749363 rad

Height: ha = 20.65767700388
Height: hb = 15.73884914582
Height: hc = 13.77111800259

Median: ma = 21.08331686423
Median: mb = 17.486570845
Median: mc = 14.33003496461

Inradius: r = 5.41881691905
Circumradius: R = 12.1999390298

Vertex coordinates: A[24; 0] B[0; 0] C[8.14658333333; 13.77111800259]
Centroid: CG[10.71552777778; 4.5990393342]
Coordinates of the circumscribed circle: U[12; 2.19766164078]
Coordinates of the inscribed circle: I[9.5; 5.41881691905]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.0221902519° = 139°1'19″ = 0.71552027222 rad
∠ B' = β' = 120.6054870564° = 120°36'18″ = 1.03766405683 rad
∠ C' = γ' = 100.3733226917° = 100°22'24″ = 1.3989749363 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 = 16 ; ; b = 21 ; ; c = 24 ; ;

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

p = a+b+c = 16+21+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 * (30.5-16)(30.5-21)(30.5-24) } ; ; T = sqrt{ 27308.94 } = 165.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 165.25 }{ 16 } = 20.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 165.25 }{ 21 } = 15.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 165.25 }{ 24 } = 13.77 ; ;

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{ 16**2-21**2-24**2 }{ 2 * 21 * 24 } ) = 40° 58'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-16**2-24**2 }{ 2 * 16 * 24 } ) = 59° 23'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-16**2-21**2 }{ 2 * 21 * 16 } ) = 79° 37'36" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 165.25 }{ 30.5 } = 5.42 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16 }{ 2 * sin 40° 58'41" } = 12.2 ; ;




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