11 12 20 triangle

Obtuse scalene triangle.

Sides: a = 11   b = 12   c = 20

Area: T = 56.71880526817
Perimeter: p = 43
Semiperimeter: s = 21.5

Angle ∠ A = α = 28.20664802662° = 28°12'23″ = 0.4922295951 rad
Angle ∠ B = β = 31.03988161007° = 31°2'20″ = 0.54217295369 rad
Angle ∠ C = γ = 120.7554703633° = 120°45'17″ = 2.10875671657 rad

Height: ha = 10.31223732148
Height: hb = 9.45330087803
Height: hc = 5.67218052682

Median: ma = 15.54883118055
Median: mb = 14.98333240638
Median: mc = 5.70108771255

Inradius: r = 2.63880489619
Circumradius: R = 11.63765066993

Vertex coordinates: A[20; 0] B[0; 0] C[9.425; 5.67218052682]
Centroid: CG[9.80883333333; 1.89106017561]
Coordinates of the circumscribed circle: U[10; -5.95504863803]
Coordinates of the inscribed circle: I[9.5; 2.63880489619]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.7943519734° = 151°47'37″ = 0.4922295951 rad
∠ B' = β' = 148.9611183899° = 148°57'40″ = 0.54217295369 rad
∠ C' = γ' = 59.2455296367° = 59°14'43″ = 2.10875671657 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 = 11 ; ; b = 12 ; ; c = 20 ; ;

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

p = a+b+c = 11+12+20 = 43 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 43 }{ 2 } = 21.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.5 * (21.5-11)(21.5-12)(21.5-20) } ; ; T = sqrt{ 3216.94 } = 56.72 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 56.72 }{ 11 } = 10.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 56.72 }{ 12 } = 9.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 56.72 }{ 20 } = 5.67 ; ;

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{ 11**2-12**2-20**2 }{ 2 * 12 * 20 } ) = 28° 12'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-11**2-20**2 }{ 2 * 11 * 20 } ) = 31° 2'20" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-11**2-12**2 }{ 2 * 12 * 11 } ) = 120° 45'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 56.72 }{ 21.5 } = 2.64 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 28° 12'23" } = 11.64 ; ;




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