11 20 24 triangle

Obtuse scalene triangle.

Sides: a = 11   b = 20   c = 24

Area: T = 109.1377241581
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 27.04881105464° = 27°2'53″ = 0.47220785855 rad
Angle ∠ B = β = 55.77111336722° = 55°46'16″ = 0.97333899101 rad
Angle ∠ C = γ = 97.18107557815° = 97°10'51″ = 1.6966124158 rad

Height: ha = 19.8433134833
Height: hb = 10.91437241581
Height: hc = 9.09547701318

Median: ma = 21.39550928953
Median: mb = 15.76438827704
Median: mc = 10.79435165725

Inradius: r = 3.96986269666
Circumradius: R = 12.09548631363

Vertex coordinates: A[24; 0] B[0; 0] C[6.18875; 9.09547701318]
Centroid: CG[10.06325; 3.03215900439]
Coordinates of the circumscribed circle: U[12; -1.5121857892]
Coordinates of the inscribed circle: I[7.5; 3.96986269666]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.9521889454° = 152°57'7″ = 0.47220785855 rad
∠ B' = β' = 124.2298866328° = 124°13'44″ = 0.97333899101 rad
∠ C' = γ' = 82.81992442185° = 82°49'9″ = 1.6966124158 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 = 20 ; ; c = 24 ; ;

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

p = a+b+c = 11+20+24 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-11)(27.5-20)(27.5-24) } ; ; T = sqrt{ 11910.94 } = 109.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 109.14 }{ 11 } = 19.84 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 109.14 }{ 20 } = 10.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 109.14 }{ 24 } = 9.09 ; ;

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**2-20**2-24**2 }{ 2 * 20 * 24 } ) = 27° 2'53" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-11**2-24**2 }{ 2 * 11 * 24 } ) = 55° 46'16" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-11**2-20**2 }{ 2 * 20 * 11 } ) = 97° 10'51" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 109.14 }{ 27.5 } = 3.97 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 27° 2'53" } = 12.09 ; ;




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