9 17 21 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 17   c = 21

Area: T = 74.41222805725
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 24.63774826477° = 24°38'15″ = 0.43300051916 rad
Angle ∠ B = β = 51.94661286461° = 51°56'46″ = 0.90766309785 rad
Angle ∠ C = γ = 103.4166388706° = 103°24'59″ = 1.80549564834 rad

Height: ha = 16.53660623494
Height: hb = 8.75443859497
Height: hc = 7.0876883864

Median: ma = 18.56774446276
Median: mb = 13.7398631664
Median: mc = 8.64658082329

Inradius: r = 3.16664800244
Circumradius: R = 10.79545891971

Vertex coordinates: A[21; 0] B[0; 0] C[5.54876190476; 7.0876883864]
Centroid: CG[8.84992063492; 2.36222946213]
Coordinates of the circumscribed circle: U[10.5; -2.50546269052]
Coordinates of the inscribed circle: I[6.5; 3.16664800244]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.3632517352° = 155°21'45″ = 0.43300051916 rad
∠ B' = β' = 128.0543871354° = 128°3'14″ = 0.90766309785 rad
∠ C' = γ' = 76.58436112938° = 76°35'1″ = 1.80549564834 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 = 9 ; ; b = 17 ; ; c = 21 ; ;

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

p = a+b+c = 9+17+21 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-9)(23.5-17)(23.5-21) } ; ; T = sqrt{ 5537.19 } = 74.41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 74.41 }{ 9 } = 16.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 74.41 }{ 17 } = 8.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 74.41 }{ 21 } = 7.09 ; ;

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{ 9**2-17**2-21**2 }{ 2 * 17 * 21 } ) = 24° 38'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-9**2-21**2 }{ 2 * 9 * 21 } ) = 51° 56'46" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-9**2-17**2 }{ 2 * 17 * 9 } ) = 103° 24'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 74.41 }{ 23.5 } = 3.17 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 24° 38'15" } = 10.79 ; ;




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