11 23 23 triangle

Acute isosceles triangle.

Sides: a = 11   b = 23   c = 23

Area: T = 122.8329912888
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 27.67704472281° = 27°40'14″ = 0.48329404096 rad
Angle ∠ B = β = 76.1654776386° = 76°9'53″ = 1.3299326122 rad
Angle ∠ C = γ = 76.1654776386° = 76°9'53″ = 1.3299326122 rad

Height: ha = 22.33327114341
Height: hb = 10.68108619902
Height: hc = 10.68108619902

Median: ma = 22.33327114341
Median: mb = 13.88334433769
Median: mc = 13.88334433769

Inradius: r = 4.31098215048
Circumradius: R = 11.84436133821

Vertex coordinates: A[23; 0] B[0; 0] C[2.63304347826; 10.68108619902]
Centroid: CG[8.54334782609; 3.56602873301]
Coordinates of the circumscribed circle: U[11.5; 2.83221684175]
Coordinates of the inscribed circle: I[5.5; 4.31098215048]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.3329552772° = 152°19'46″ = 0.48329404096 rad
∠ B' = β' = 103.8355223614° = 103°50'7″ = 1.3299326122 rad
∠ C' = γ' = 103.8355223614° = 103°50'7″ = 1.3299326122 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 = 23 ; ; c = 23 ; ;

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

p = a+b+c = 11+23+23 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-11)(28.5-23)(28.5-23) } ; ; T = sqrt{ 15087.19 } = 122.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 122.83 }{ 11 } = 22.33 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 122.83 }{ 23 } = 10.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 122.83 }{ 23 } = 10.68 ; ;

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**2-23**2-23**2 }{ 2 * 23 * 23 } ) = 27° 40'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-11**2-23**2 }{ 2 * 11 * 23 } ) = 76° 9'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-11**2-23**2 }{ 2 * 23 * 11 } ) = 76° 9'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 122.83 }{ 28.5 } = 4.31 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 27° 40'14" } = 11.84 ; ;




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