11 20 21 triangle

Acute scalene triangle.

Sides: a = 11   b = 20   c = 21

Area: T = 108.1676538264
Perimeter: p = 52
Semiperimeter: s = 26

Angle ∠ A = α = 31.00327191339° = 31°10″ = 0.5411099526 rad
Angle ∠ B = β = 69.47329625625° = 69°28'23″ = 1.21325319378 rad
Angle ∠ C = γ = 79.52443183036° = 79°31'28″ = 1.38879611898 rad

Height: ha = 19.66766433207
Height: hb = 10.81766538264
Height: hc = 10.30215750728

Median: ma = 19.75547462651
Median: mb = 13.45436240471
Median: mc = 12.25876506721

Inradius: r = 4.16602514717
Circumradius: R = 10.67879787773

Vertex coordinates: A[21; 0] B[0; 0] C[3.85771428571; 10.30215750728]
Centroid: CG[8.28657142857; 3.43438583576]
Coordinates of the circumscribed circle: U[10.5; 1.94114506868]
Coordinates of the inscribed circle: I[6; 4.16602514717]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.9977280866° = 148°59'50″ = 0.5411099526 rad
∠ B' = β' = 110.5277037437° = 110°31'37″ = 1.21325319378 rad
∠ C' = γ' = 100.4765681696° = 100°28'32″ = 1.38879611898 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 = 20 ; ; c = 21 ; ;

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

p = a+b+c = 11+20+21 = 52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 52 }{ 2 } = 26 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26 * (26-11)(26-20)(26-21) } ; ; T = sqrt{ 11700 } = 108.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 108.17 }{ 11 } = 19.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 108.17 }{ 20 } = 10.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 108.17 }{ 21 } = 10.3 ; ;

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**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 31° 10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-11**2-21**2 }{ 2 * 11 * 21 } ) = 69° 28'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-11**2-20**2 }{ 2 * 20 * 11 } ) = 79° 31'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 108.17 }{ 26 } = 4.16 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 31° 10" } = 10.68 ; ;




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