20 21 24 triangle

Acute scalene triangle.

Sides: a = 20   b = 21   c = 24

Area: T = 199.2766033431
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 52.25882679659° = 52°15'30″ = 0.91220788374 rad
Angle ∠ B = β = 56.13112953983° = 56°7'53″ = 0.98796759181 rad
Angle ∠ C = γ = 71.61104366358° = 71°36'38″ = 1.25498378981 rad

Height: ha = 19.92876033431
Height: hb = 18.97986698506
Height: hc = 16.60663361193

Median: ma = 20.21113829314
Median: mb = 19.43657917256
Median: mc = 16.62882891483

Inradius: r = 6.13215702594
Circumradius: R = 12.64657755938

Vertex coordinates: A[24; 0] B[0; 0] C[11.14658333333; 16.60663361193]
Centroid: CG[11.71552777778; 5.53554453731]
Coordinates of the circumscribed circle: U[12; 3.989944111]
Coordinates of the inscribed circle: I[11.5; 6.13215702594]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.7421732034° = 127°44'30″ = 0.91220788374 rad
∠ B' = β' = 123.8698704602° = 123°52'7″ = 0.98796759181 rad
∠ C' = γ' = 108.3989563364° = 108°23'22″ = 1.25498378981 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 = 24 ; ;

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

p = a+b+c = 20+21+24 = 65 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 65 }{ 2 } = 32.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.5 * (32.5-20)(32.5-21)(32.5-24) } ; ; T = sqrt{ 39710.94 } = 199.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 199.28 }{ 20 } = 19.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 199.28 }{ 21 } = 18.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 199.28 }{ 24 } = 16.61 ; ;

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**2-24**2 }{ 2 * 21 * 24 } ) = 52° 15'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-20**2-24**2 }{ 2 * 20 * 24 } ) = 56° 7'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-20**2-21**2 }{ 2 * 21 * 20 } ) = 71° 36'38" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 199.28 }{ 32.5 } = 6.13 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 52° 15'30" } = 12.65 ; ;




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