21 21 22 triangle

Acute isosceles triangle.

Sides: a = 21   b = 21   c = 22

Area: T = 196.774398202
Perimeter: p = 64
Semiperimeter: s = 32

Angle ∠ A = α = 58.41218644948° = 58°24'43″ = 1.01994793577 rad
Angle ∠ B = β = 58.41218644948° = 58°24'43″ = 1.01994793577 rad
Angle ∠ C = γ = 63.17662710104° = 63°10'35″ = 1.10326339383 rad

Height: ha = 18.744037924
Height: hb = 18.744037924
Height: hc = 17.889854382

Median: ma = 18.76883243791
Median: mb = 18.76883243791
Median: mc = 17.889854382

Inradius: r = 6.14991869381
Circumradius: R = 12.3266324726

Vertex coordinates: A[22; 0] B[0; 0] C[11; 17.889854382]
Centroid: CG[11; 5.963284794]
Coordinates of the circumscribed circle: U[11; 5.5622219094]
Coordinates of the inscribed circle: I[11; 6.14991869381]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.5888135505° = 121°35'17″ = 1.01994793577 rad
∠ B' = β' = 121.5888135505° = 121°35'17″ = 1.01994793577 rad
∠ C' = γ' = 116.824372899° = 116°49'25″ = 1.10326339383 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 = 21 ; ; b = 21 ; ; c = 22 ; ;

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

p = a+b+c = 21+21+22 = 64 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 64 }{ 2 } = 32 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32 * (32-21)(32-21)(32-22) } ; ; T = sqrt{ 38720 } = 196.77 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 196.77 }{ 21 } = 18.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 196.77 }{ 21 } = 18.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 196.77 }{ 22 } = 17.89 ; ;

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{ 21**2-21**2-22**2 }{ 2 * 21 * 22 } ) = 58° 24'43" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-21**2-22**2 }{ 2 * 21 * 22 } ) = 58° 24'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-21**2-21**2 }{ 2 * 21 * 21 } ) = 63° 10'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 196.77 }{ 32 } = 6.15 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 58° 24'43" } = 12.33 ; ;




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