16 18 21 triangle

Acute scalene triangle.

Sides: a = 16   b = 18   c = 21

Area: T = 139.7444185926
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 47.67992428549° = 47°40'45″ = 0.83221597727 rad
Angle ∠ B = β = 56.2855185269° = 56°17'7″ = 0.9822361803 rad
Angle ∠ C = γ = 76.03655718761° = 76°2'8″ = 1.32770710779 rad

Height: ha = 17.46880232407
Height: hb = 15.52771317695
Height: hc = 13.30989700881

Median: ma = 17.84765682976
Median: mb = 16.35554272338
Median: mc = 13.40770876778

Inradius: r = 5.08216067609
Circumradius: R = 10.82197703538

Vertex coordinates: A[21; 0] B[0; 0] C[8.8810952381; 13.30989700881]
Centroid: CG[9.96603174603; 4.43663233627]
Coordinates of the circumscribed circle: U[10.5; 2.61110209708]
Coordinates of the inscribed circle: I[9.5; 5.08216067609]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.3210757145° = 132°19'15″ = 0.83221597727 rad
∠ B' = β' = 123.7154814731° = 123°42'53″ = 0.9822361803 rad
∠ C' = γ' = 103.9644428124° = 103°57'52″ = 1.32770710779 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 = 18 ; ; c = 21 ; ;

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

p = a+b+c = 16+18+21 = 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 * (27.5-16)(27.5-18)(27.5-21) } ; ; T = sqrt{ 19528.44 } = 139.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 139.74 }{ 16 } = 17.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 139.74 }{ 18 } = 15.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 139.74 }{ 21 } = 13.31 ; ;

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**2-18**2-21**2 }{ 2 * 18 * 21 } ) = 47° 40'45" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-16**2-21**2 }{ 2 * 16 * 21 } ) = 56° 17'7" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-16**2-18**2 }{ 2 * 18 * 16 } ) = 76° 2'8" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 139.74 }{ 27.5 } = 5.08 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16 }{ 2 * sin 47° 40'45" } = 10.82 ; ;




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