11 21 23 triangle

Acute scalene triangle.

Sides: a = 11   b = 21   c = 23

Area: T = 115.2054980361
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 28.49222090409° = 28°29'32″ = 0.49772828589 rad
Angle ∠ B = β = 65.60438347173° = 65°36'14″ = 1.14550029178 rad
Angle ∠ C = γ = 85.90439562418° = 85°54'14″ = 1.49993068769 rad

Height: ha = 20.94663600657
Height: hb = 10.97219028915
Height: hc = 10.01878243792

Median: ma = 21.32548681121
Median: mb = 14.65443508898
Median: mc = 12.19663109177

Inradius: r = 4.18992720131
Circumradius: R = 11.5299449472

Vertex coordinates: A[23; 0] B[0; 0] C[4.54334782609; 10.01878243792]
Centroid: CG[9.18111594203; 3.33992747931]
Coordinates of the circumscribed circle: U[11.5; 0.82435321051]
Coordinates of the inscribed circle: I[6.5; 4.18992720131]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.5087790959° = 151°30'28″ = 0.49772828589 rad
∠ B' = β' = 114.3966165283° = 114°23'46″ = 1.14550029178 rad
∠ C' = γ' = 94.09660437582° = 94°5'46″ = 1.49993068769 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 = 21 ; ; c = 23 ; ;

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

p = a+b+c = 11+21+23 = 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)(27.5-21)(27.5-23) } ; ; T = sqrt{ 13272.19 } = 115.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 115.2 }{ 11 } = 20.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 115.2 }{ 21 } = 10.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 115.2 }{ 23 } = 10.02 ; ;

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**2-21**2-23**2 }{ 2 * 21 * 23 } ) = 28° 29'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-11**2-23**2 }{ 2 * 11 * 23 } ) = 65° 36'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-11**2-21**2 }{ 2 * 21 * 11 } ) = 85° 54'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 115.2 }{ 27.5 } = 4.19 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 28° 29'32" } = 11.53 ; ;




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