17 21 22 triangle

Acute scalene triangle.

Sides: a = 17   b = 21   c = 22

Area: T = 167.5710880525
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 46.50333874881° = 46°30'12″ = 0.8121637225 rad
Angle ∠ B = β = 63.65501973562° = 63°39'1″ = 1.11109055134 rad
Angle ∠ C = γ = 69.84664151556° = 69°50'47″ = 1.21990499152 rad

Height: ha = 19.71442212383
Height: hb = 15.95991314786
Height: hc = 15.23437164114

Median: ma = 19.75547462651
Median: mb = 16.62107701386
Median: mc = 15.62204993518

Inradius: r = 5.58656960175
Circumradius: R = 11.71774296265

Vertex coordinates: A[22; 0] B[0; 0] C[7.54554545455; 15.23437164114]
Centroid: CG[9.84884848485; 5.07879054705]
Coordinates of the circumscribed circle: U[11; 4.03770976024]
Coordinates of the inscribed circle: I[9; 5.58656960175]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.4976612512° = 133°29'48″ = 0.8121637225 rad
∠ B' = β' = 116.3549802644° = 116°20'59″ = 1.11109055134 rad
∠ C' = γ' = 110.1543584844° = 110°9'13″ = 1.21990499152 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 = 21 ; ; c = 22 ; ;

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

p = a+b+c = 17+21+22 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-17)(30-21)(30-22) } ; ; T = sqrt{ 28080 } = 167.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 167.57 }{ 17 } = 19.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 167.57 }{ 21 } = 15.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 167.57 }{ 22 } = 15.23 ; ;

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**2-21**2-22**2 }{ 2 * 21 * 22 } ) = 46° 30'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-17**2-22**2 }{ 2 * 17 * 22 } ) = 63° 39'1" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-17**2-21**2 }{ 2 * 21 * 17 } ) = 69° 50'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 167.57 }{ 30 } = 5.59 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 17 }{ 2 * sin 46° 30'12" } = 11.72 ; ;




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