16 17 21 triangle

Acute scalene triangle.

Sides: a = 16   b = 17   c = 21

Area: T = 133.4921572768
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 48.40546480534° = 48°24'17″ = 0.84548204818 rad
Angle ∠ B = β = 52.61768015821° = 52°37' = 0.91883364295 rad
Angle ∠ C = γ = 78.97985503645° = 78°58'43″ = 1.37884357423 rad

Height: ha = 16.6866446596
Height: hb = 15.70548909138
Height: hc = 12.71334831207

Median: ma = 17.34993515729
Median: mb = 16.62107701386
Median: mc = 12.73877392029

Inradius: r = 4.94441323247
Circumradius: R = 10.69773044844

Vertex coordinates: A[21; 0] B[0; 0] C[9.71442857143; 12.71334831207]
Centroid: CG[10.23880952381; 4.23878277069]
Coordinates of the circumscribed circle: U[10.5; 2.04550729161]
Coordinates of the inscribed circle: I[10; 4.94441323247]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.5955351947° = 131°35'43″ = 0.84548204818 rad
∠ B' = β' = 127.3833198418° = 127°23' = 0.91883364295 rad
∠ C' = γ' = 101.0211449636° = 101°1'17″ = 1.37884357423 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 = 17 ; ; c = 21 ; ;

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

p = a+b+c = 16+17+21 = 54 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54 }{ 2 } = 27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27 * (27-16)(27-17)(27-21) } ; ; T = sqrt{ 17820 } = 133.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 133.49 }{ 16 } = 16.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 133.49 }{ 17 } = 15.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 133.49 }{ 21 } = 12.71 ; ;

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**2-17**2-21**2 }{ 2 * 17 * 21 } ) = 48° 24'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-16**2-21**2 }{ 2 * 16 * 21 } ) = 52° 37' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-16**2-17**2 }{ 2 * 17 * 16 } ) = 78° 58'43" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 133.49 }{ 27 } = 4.94 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16 }{ 2 * sin 48° 24'17" } = 10.7 ; ;




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