10 11 11 triangle

Acute isosceles triangle.

Sides: a = 10   b = 11   c = 11

Area: T = 48.99897948557
Perimeter: p = 32
Semiperimeter: s = 16

Angle ∠ A = α = 54.07113835788° = 54°4'17″ = 0.94437236746 rad
Angle ∠ B = β = 62.96443082106° = 62°57'52″ = 1.09989344895 rad
Angle ∠ C = γ = 62.96443082106° = 62°57'52″ = 1.09989344895 rad

Height: ha = 9.79879589711
Height: hb = 8.90772354283
Height: hc = 8.90772354283

Median: ma = 9.79879589711
Median: mb = 8.95882364336
Median: mc = 8.95882364336

Inradius: r = 3.06218621785
Circumradius: R = 6.17547553933

Vertex coordinates: A[11; 0] B[0; 0] C[4.54554545455; 8.90772354283]
Centroid: CG[5.18218181818; 2.96990784761]
Coordinates of the circumscribed circle: U[5.5; 2.80767069969]
Coordinates of the inscribed circle: I[5; 3.06218621785]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.9298616421° = 125°55'43″ = 0.94437236746 rad
∠ B' = β' = 117.0365691789° = 117°2'8″ = 1.09989344895 rad
∠ C' = γ' = 117.0365691789° = 117°2'8″ = 1.09989344895 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 = 11 ; ;

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

p = a+b+c = 10+11+11 = 32 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 32 }{ 2 } = 16 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16 * (16-10)(16-11)(16-11) } ; ; T = sqrt{ 2400 } = 48.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 48.99 }{ 10 } = 9.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 48.99 }{ 11 } = 8.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 48.99 }{ 11 } = 8.91 ; ;

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**2-11**2 }{ 2 * 11 * 11 } ) = 54° 4'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11**2-10**2-11**2 }{ 2 * 10 * 11 } ) = 62° 57'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11**2-10**2-11**2 }{ 2 * 11 * 10 } ) = 62° 57'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 48.99 }{ 16 } = 3.06 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 54° 4'17" } = 6.17 ; ;




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