10 11 20 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 11   c = 20

Area: T = 31.97655766172
Perimeter: p = 41
Semiperimeter: s = 20.5

Angle ∠ A = α = 16.89990937835° = 16°53'57″ = 0.29549448271 rad
Angle ∠ B = β = 18.64881553056° = 18°38'53″ = 0.32554717095 rad
Angle ∠ C = γ = 144.4532750911° = 144°27'10″ = 2.5211176117 rad

Height: ha = 6.39551153234
Height: hb = 5.81437412031
Height: hc = 3.19875576617

Median: ma = 15.34660092532
Median: mb = 14.82439670804
Median: mc = 3.24403703492

Inradius: r = 1.56597842252
Circumradius: R = 17.20106280476

Vertex coordinates: A[20; 0] B[0; 0] C[9.475; 3.19875576617]
Centroid: CG[9.825; 1.06658525539]
Coordinates of the circumscribed circle: U[10; -13.99550564569]
Coordinates of the inscribed circle: I[9.5; 1.56597842252]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163.1010906217° = 163°6'3″ = 0.29549448271 rad
∠ B' = β' = 161.3521844694° = 161°21'7″ = 0.32554717095 rad
∠ C' = γ' = 35.54772490891° = 35°32'50″ = 2.5211176117 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 = 10 ; ; b = 11 ; ; c = 20 ; ;

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

p = a+b+c = 10+11+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 * (20.5-10)(20.5-11)(20.5-20) } ; ; T = sqrt{ 1022.44 } = 31.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 31.98 }{ 10 } = 6.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 31.98 }{ 11 } = 5.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 31.98 }{ 20 } = 3.2 ; ;

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{ 10**2-11**2-20**2 }{ 2 * 11 * 20 } ) = 16° 53'57" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11**2-10**2-20**2 }{ 2 * 10 * 20 } ) = 18° 38'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-10**2-11**2 }{ 2 * 11 * 10 } ) = 144° 27'10" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 31.98 }{ 20.5 } = 1.56 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 16° 53'57" } = 17.2 ; ;




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