20 21 21 triangle

Acute isosceles triangle.

Sides: a = 20   b = 21   c = 21

Area: T = 184.6621853126
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 56.87437802977° = 56°52'26″ = 0.99326347243 rad
Angle ∠ B = β = 61.56331098511° = 61°33'47″ = 1.07444789647 rad
Angle ∠ C = γ = 61.56331098511° = 61°33'47″ = 1.07444789647 rad

Height: ha = 18.46661853126
Height: hb = 17.58768431549
Height: hc = 17.58768431549

Median: ma = 18.46661853126
Median: mb = 17.61439149538
Median: mc = 17.61439149538

Inradius: r = 5.95768339718
Circumradius: R = 11.94107444617

Vertex coordinates: A[21; 0] B[0; 0] C[9.52438095238; 17.58768431549]
Centroid: CG[10.17546031746; 5.86222810516]
Coordinates of the circumscribed circle: U[10.5; 5.68660687913]
Coordinates of the inscribed circle: I[10; 5.95768339718]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.1266219702° = 123°7'34″ = 0.99326347243 rad
∠ B' = β' = 118.4376890149° = 118°26'13″ = 1.07444789647 rad
∠ C' = γ' = 118.4376890149° = 118°26'13″ = 1.07444789647 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 = 20 ; ; b = 21 ; ; c = 21 ; ;

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

p = a+b+c = 20+21+21 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-20)(31-21)(31-21) } ; ; T = sqrt{ 34100 } = 184.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 184.66 }{ 20 } = 18.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 184.66 }{ 21 } = 17.59 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 184.66 }{ 21 } = 17.59 ; ;

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{ 20**2-21**2-21**2 }{ 2 * 21 * 21 } ) = 56° 52'26" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 61° 33'47" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-20**2-21**2 }{ 2 * 21 * 20 } ) = 61° 33'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 184.66 }{ 31 } = 5.96 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 56° 52'26" } = 11.94 ; ;




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