20 27 29 triangle

Acute scalene triangle.

Sides: a = 20   b = 27   c = 29

Area: T = 260.2232981306
Perimeter: p = 76
Semiperimeter: s = 38

Angle ∠ A = α = 41.65879296759° = 41°39'29″ = 0.72770680324 rad
Angle ∠ B = β = 63.80880802775° = 63°48'29″ = 1.11436610902 rad
Angle ∠ C = γ = 74.53439900466° = 74°32'2″ = 1.3010863531 rad

Height: ha = 26.02222981306
Height: hb = 19.27657763931
Height: hc = 17.94664125039

Median: ma = 26.17325046566
Median: mb = 20.93444214155
Median: mc = 18.82215302247

Inradius: r = 6.84879731923
Circumradius: R = 15.04547895891

Vertex coordinates: A[29; 0] B[0; 0] C[8.82875862069; 17.94664125039]
Centroid: CG[12.60991954023; 5.98221375013]
Coordinates of the circumscribed circle: U[14.5; 4.01219438904]
Coordinates of the inscribed circle: I[11; 6.84879731923]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.3422070324° = 138°20'31″ = 0.72770680324 rad
∠ B' = β' = 116.1921919723° = 116°11'31″ = 1.11436610902 rad
∠ C' = γ' = 105.4666009953° = 105°27'58″ = 1.3010863531 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 = 20 ; ; b = 27 ; ; c = 29 ; ;

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

p = a+b+c = 20+27+29 = 76 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 76 }{ 2 } = 38 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38 * (38-20)(38-27)(38-29) } ; ; T = sqrt{ 67716 } = 260.22 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 260.22 }{ 20 } = 26.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 260.22 }{ 27 } = 19.28 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 260.22 }{ 29 } = 17.95 ; ;

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{ 20**2-27**2-29**2 }{ 2 * 27 * 29 } ) = 41° 39'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-20**2-29**2 }{ 2 * 20 * 29 } ) = 63° 48'29" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-20**2-27**2 }{ 2 * 27 * 20 } ) = 74° 32'2" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 260.22 }{ 38 } = 6.85 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 41° 39'29" } = 15.04 ; ;




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