5 18 21 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 18   c = 21

Area: T = 38.67881592116
Perimeter: p = 44
Semiperimeter: s = 22

Angle ∠ A = α = 11.80987972119° = 11°48'32″ = 0.2066102392 rad
Angle ∠ B = β = 47.4533334116° = 47°27'12″ = 0.82882169214 rad
Angle ∠ C = γ = 120.7387868672° = 120°44'16″ = 2.10772733402 rad

Height: ha = 15.47112636847
Height: hb = 4.29875732457
Height: hc = 3.68436342106

Median: ma = 19.39771647413
Median: mb = 12.32988280059
Median: mc = 8.01656097709

Inradius: r = 1.7588098146
Circumradius: R = 12.21661966761

Vertex coordinates: A[21; 0] B[0; 0] C[3.3810952381; 3.68436342106]
Centroid: CG[8.1276984127; 1.22878780702]
Coordinates of the circumscribed circle: U[10.5; -6.24438338567]
Coordinates of the inscribed circle: I[4; 1.7588098146]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 168.1911202788° = 168°11'28″ = 0.2066102392 rad
∠ B' = β' = 132.5476665884° = 132°32'48″ = 0.82882169214 rad
∠ C' = γ' = 59.26221313279° = 59°15'44″ = 2.10772733402 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 = 5 ; ; b = 18 ; ; c = 21 ; ;

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

p = a+b+c = 5+18+21 = 44 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 44 }{ 2 } = 22 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22 * (22-5)(22-18)(22-21) } ; ; T = sqrt{ 1496 } = 38.68 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 38.68 }{ 5 } = 15.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 38.68 }{ 18 } = 4.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 38.68 }{ 21 } = 3.68 ; ;

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{ 5**2-18**2-21**2 }{ 2 * 18 * 21 } ) = 11° 48'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-5**2-21**2 }{ 2 * 5 * 21 } ) = 47° 27'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-5**2-18**2 }{ 2 * 18 * 5 } ) = 120° 44'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 38.68 }{ 22 } = 1.76 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 11° 48'32" } = 12.22 ; ;




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