9 12 20 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 12   c = 20

Area: T = 31.65333963423
Perimeter: p = 41
Semiperimeter: s = 20.5

Angle ∠ A = α = 15.29443719921° = 15°17'40″ = 0.26769371483 rad
Angle ∠ B = β = 20.59215995516° = 20°35'30″ = 0.35993912104 rad
Angle ∠ C = γ = 144.1144028456° = 144°6'51″ = 2.51552642949 rad

Height: ha = 7.03440880761
Height: hb = 5.2765566057
Height: hc = 3.16553396342

Median: ma = 15.86766316526
Median: mb = 14.33003496461
Median: mc = 3.53655339059

Inradius: r = 1.54440681143
Circumradius: R = 17.06597806997

Vertex coordinates: A[20; 0] B[0; 0] C[8.425; 3.16553396342]
Centroid: CG[9.475; 1.05551132114]
Coordinates of the circumscribed circle: U[10; -13.82215815854]
Coordinates of the inscribed circle: I[8.5; 1.54440681143]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 164.7065628008° = 164°42'20″ = 0.26769371483 rad
∠ B' = β' = 159.4088400448° = 159°24'30″ = 0.35993912104 rad
∠ C' = γ' = 35.88659715437° = 35°53'9″ = 2.51552642949 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 = 12 ; ; c = 20 ; ;

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

p = a+b+c = 9+12+20 = 41 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 41 }{ 2 } = 20.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.5 * (20.5-9)(20.5-12)(20.5-20) } ; ; T = sqrt{ 1001.94 } = 31.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 31.65 }{ 9 } = 7.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 31.65 }{ 12 } = 5.28 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 31.65 }{ 20 } = 3.17 ; ;

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**2-12**2-20**2 }{ 2 * 12 * 20 } ) = 15° 17'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-9**2-20**2 }{ 2 * 9 * 20 } ) = 20° 35'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-9**2-12**2 }{ 2 * 12 * 9 } ) = 144° 6'51" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 31.65 }{ 20.5 } = 1.54 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 15° 17'40" } = 17.06 ; ;




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