14 16 21 triangle

Acute scalene triangle.

Sides: a = 14   b = 16   c = 21

Area: T = 111.9666233749
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 41.79548666174° = 41°47'42″ = 0.72994580329 rad
Angle ∠ B = β = 49.61220907921° = 49°36'44″ = 0.86658943331 rad
Angle ∠ C = γ = 88.59330425904° = 88°35'35″ = 1.54662402876 rad

Height: ha = 15.99551762499
Height: hb = 13.99657792187
Height: hc = 10.66334508333

Median: ma = 17.30660682999
Median: mb = 15.95330561335
Median: mc = 10.75987173957

Inradius: r = 4.39108326961
Circumradius: R = 10.50331665407

Vertex coordinates: A[21; 0] B[0; 0] C[9.07114285714; 10.66334508333]
Centroid: CG[10.02438095238; 3.55444836111]
Coordinates of the circumscribed circle: U[10.5; 0.25878902499]
Coordinates of the inscribed circle: I[9.5; 4.39108326961]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.2055133383° = 138°12'18″ = 0.72994580329 rad
∠ B' = β' = 130.3887909208° = 130°23'16″ = 0.86658943331 rad
∠ C' = γ' = 91.40769574096° = 91°24'25″ = 1.54662402876 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 = 14 ; ; b = 16 ; ; c = 21 ; ;

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

p = a+b+c = 14+16+21 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-14)(25.5-16)(25.5-21) } ; ; T = sqrt{ 12536.44 } = 111.97 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 111.97 }{ 14 } = 16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 111.97 }{ 16 } = 14 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 111.97 }{ 21 } = 10.66 ; ;

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{ 14**2-16**2-21**2 }{ 2 * 16 * 21 } ) = 41° 47'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-14**2-21**2 }{ 2 * 14 * 21 } ) = 49° 36'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-14**2-16**2 }{ 2 * 16 * 14 } ) = 88° 35'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 111.97 }{ 25.5 } = 4.39 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 41° 47'42" } = 10.5 ; ;




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