20 20 21 triangle

Acute isosceles triangle.

Sides: a = 20   b = 20   c = 21

Area: T = 178.7311467571
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 58.33217567465° = 58°19'54″ = 1.01880812137 rad
Angle ∠ B = β = 58.33217567465° = 58°19'54″ = 1.01880812137 rad
Angle ∠ C = γ = 63.33664865071° = 63°20'11″ = 1.10554302262 rad

Height: ha = 17.87331467571
Height: hb = 17.87331467571
Height: hc = 17.02220445305

Median: ma = 17.903251379
Median: mb = 17.903251379
Median: mc = 17.02220445305

Inradius: r = 5.86600481171
Circumradius: R = 11.74994699089

Vertex coordinates: A[21; 0] B[0; 0] C[10.5; 17.02220445305]
Centroid: CG[10.5; 5.67440148435]
Coordinates of the circumscribed circle: U[10.5; 5.27325746216]
Coordinates of the inscribed circle: I[10.5; 5.86600481171]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.6688243254° = 121°40'6″ = 1.01880812137 rad
∠ B' = β' = 121.6688243254° = 121°40'6″ = 1.01880812137 rad
∠ C' = γ' = 116.6643513493° = 116°39'49″ = 1.10554302262 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 = 20 ; ; c = 21 ; ;

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

p = a+b+c = 20+20+21 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-20)(30.5-20)(30.5-21) } ; ; T = sqrt{ 31944.94 } = 178.73 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 178.73 }{ 20 } = 17.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 178.73 }{ 20 } = 17.87 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 178.73 }{ 21 } = 17.02 ; ;

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**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 58° 19'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 58° 19'54" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-20**2-20**2 }{ 2 * 20 * 20 } ) = 63° 20'11" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 178.73 }{ 30.5 } = 5.86 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 58° 19'54" } = 11.75 ; ;




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