13 18 20 triangle

Acute scalene triangle.

Sides: a = 13   b = 18   c = 20

Area: T = 114.6676636386
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 39.57112194572° = 39°34'16″ = 0.69106480686 rad
Angle ∠ B = β = 61.89107787174° = 61°53'27″ = 1.08801978652 rad
Angle ∠ C = γ = 78.53880018254° = 78°32'17″ = 1.37107467198 rad

Height: ha = 17.64110209824
Height: hb = 12.74107373762
Height: hc = 11.46766636386

Median: ma = 17.88215547422
Median: mb = 14.26553426177
Median: mc = 12.10437184369

Inradius: r = 4.49767308387
Circumradius: R = 10.20334910666

Vertex coordinates: A[20; 0] B[0; 0] C[6.125; 11.46766636386]
Centroid: CG[8.70883333333; 3.82222212129]
Coordinates of the circumscribed circle: U[10; 2.02876168145]
Coordinates of the inscribed circle: I[7.5; 4.49767308387]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.4298780543° = 140°25'44″ = 0.69106480686 rad
∠ B' = β' = 118.1099221283° = 118°6'33″ = 1.08801978652 rad
∠ C' = γ' = 101.4621998175° = 101°27'43″ = 1.37107467198 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 = 13 ; ; b = 18 ; ; c = 20 ; ;

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

p = a+b+c = 13+18+20 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-13)(25.5-18)(25.5-20) } ; ; T = sqrt{ 13148.44 } = 114.67 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 114.67 }{ 13 } = 17.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 114.67 }{ 18 } = 12.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 114.67 }{ 20 } = 11.47 ; ;

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{ 13**2-18**2-20**2 }{ 2 * 18 * 20 } ) = 39° 34'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-13**2-20**2 }{ 2 * 13 * 20 } ) = 61° 53'27" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-13**2-18**2 }{ 2 * 18 * 13 } ) = 78° 32'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 114.67 }{ 25.5 } = 4.5 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 39° 34'16" } = 10.2 ; ;




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