17 18 21 triangle

Acute scalene triangle.

Sides: a = 17   b = 18   c = 21

Area: T = 146.8333238744
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 50.97771974348° = 50°58'38″ = 0.89897199387 rad
Angle ∠ B = β = 55.34554309073° = 55°20'44″ = 0.96659599953 rad
Angle ∠ C = γ = 73.6777371658° = 73°40'39″ = 1.28659127196 rad

Height: ha = 17.27444986757
Height: hb = 16.31548043049
Height: hc = 13.98441179756

Median: ma = 17.61439149538
Median: mb = 16.85222995464
Median: mc = 14.00989257261

Inradius: r = 5.24440442409
Circumradius: R = 10.94109832116

Vertex coordinates: A[21; 0] B[0; 0] C[9.66766666667; 13.98441179756]
Centroid: CG[10.22222222222; 4.66113726585]
Coordinates of the circumscribed circle: U[10.5; 3.07549168503]
Coordinates of the inscribed circle: I[10; 5.24440442409]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.0232802565° = 129°1'22″ = 0.89897199387 rad
∠ B' = β' = 124.6554569093° = 124°39'16″ = 0.96659599953 rad
∠ C' = γ' = 106.3232628342° = 106°19'21″ = 1.28659127196 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 = 17 ; ; b = 18 ; ; c = 21 ; ;

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

p = a+b+c = 17+18+21 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-17)(28-18)(28-21) } ; ; T = sqrt{ 21560 } = 146.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 146.83 }{ 17 } = 17.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 146.83 }{ 18 } = 16.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 146.83 }{ 21 } = 13.98 ; ;

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{ 17**2-18**2-21**2 }{ 2 * 18 * 21 } ) = 50° 58'38" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-17**2-21**2 }{ 2 * 17 * 21 } ) = 55° 20'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-17**2-18**2 }{ 2 * 18 * 17 } ) = 73° 40'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 146.83 }{ 28 } = 5.24 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 17 }{ 2 * sin 50° 58'38" } = 10.94 ; ;




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