9 20 21 triangle

Acute scalene triangle.

Sides: a = 9   b = 20   c = 21

Area: T = 89.44327191
Perimeter: p = 50
Semiperimeter: s = 25

Angle ∠ A = α = 25.20987652968° = 25°12'32″ = 0.44399759548 rad
Angle ∠ B = β = 71.17106049117° = 71°10'14″ = 1.24221613863 rad
Angle ∠ C = γ = 83.62106297916° = 83°37'14″ = 1.45994553125 rad

Height: ha = 19.87661598
Height: hb = 8.944427191
Height: hc = 8.51883542

Median: ma = 20.00662490237
Median: mb = 12.68985775404
Median: mc = 11.41327122105

Inradius: r = 3.5787708764
Circumradius: R = 10.56554211937

Vertex coordinates: A[21; 0] B[0; 0] C[2.90547619048; 8.51883542]
Centroid: CG[7.96882539683; 2.83994514]
Coordinates of the circumscribed circle: U[10.5; 1.17439356882]
Coordinates of the inscribed circle: I[5; 3.5787708764]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.7911234703° = 154°47'28″ = 0.44399759548 rad
∠ B' = β' = 108.8299395088° = 108°49'46″ = 1.24221613863 rad
∠ C' = γ' = 96.37993702084° = 96°22'46″ = 1.45994553125 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 = 20 ; ; c = 21 ; ;

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

p = a+b+c = 9+20+21 = 50 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 50 }{ 2 } = 25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25 * (25-9)(25-20)(25-21) } ; ; T = sqrt{ 8000 } = 89.44 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 89.44 }{ 9 } = 19.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 89.44 }{ 20 } = 8.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 89.44 }{ 21 } = 8.52 ; ;

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**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 25° 12'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-9**2-21**2 }{ 2 * 9 * 21 } ) = 71° 10'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-9**2-20**2 }{ 2 * 20 * 9 } ) = 83° 37'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 89.44 }{ 25 } = 3.58 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 25° 12'32" } = 10.57 ; ;




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