11 21 21 triangle

Acute isosceles triangle.

Sides: a = 11   b = 21   c = 21

Area: T = 111.4688325097
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 30.36662280872° = 30°21'58″ = 0.53299906615 rad
Angle ∠ B = β = 74.81768859564° = 74°49'1″ = 1.3065800996 rad
Angle ∠ C = γ = 74.81768859564° = 74°49'1″ = 1.3065800996 rad

Height: ha = 20.26769681995
Height: hb = 10.61660309616
Height: hc = 10.61660309616

Median: ma = 20.26769681995
Median: mb = 13.06771343454
Median: mc = 13.06771343454

Inradius: r = 4.20663518905
Circumradius: R = 10.88797723384

Vertex coordinates: A[21; 0] B[0; 0] C[2.8810952381; 10.61660309616]
Centroid: CG[7.96603174603; 3.53986769872]
Coordinates of the circumscribed circle: U[10.5; 2.84994641839]
Coordinates of the inscribed circle: I[5.5; 4.20663518905]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.6343771913° = 149°38'2″ = 0.53299906615 rad
∠ B' = β' = 105.1833114044° = 105°10'59″ = 1.3065800996 rad
∠ C' = γ' = 105.1833114044° = 105°10'59″ = 1.3065800996 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 = 21 ; ;

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

p = a+b+c = 11+21+21 = 53 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53 }{ 2 } = 26.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.5 * (26.5-11)(26.5-21)(26.5-21) } ; ; T = sqrt{ 12425.19 } = 111.47 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 111.47 }{ 11 } = 20.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 111.47 }{ 21 } = 10.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 111.47 }{ 21 } = 10.62 ; ;

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**2-21**2 }{ 2 * 21 * 21 } ) = 30° 21'58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-11**2-21**2 }{ 2 * 11 * 21 } ) = 74° 49'1" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-11**2-21**2 }{ 2 * 21 * 11 } ) = 74° 49'1" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 111.47 }{ 26.5 } = 4.21 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 30° 21'58" } = 10.88 ; ;




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