20 22 25 triangle

Acute scalene triangle.

Sides: a = 20   b = 22   c = 25

Area: T = 210.2565647962
Perimeter: p = 67
Semiperimeter: s = 33.5

Angle ∠ A = α = 49.86883962361° = 49°52'6″ = 0.8770367707 rad
Angle ∠ B = β = 57.24882608451° = 57°14'54″ = 0.99991706428 rad
Angle ∠ C = γ = 72.88333429188° = 72°53' = 1.27220543038 rad

Height: ha = 21.02655647962
Height: hb = 19.11441498147
Height: hc = 16.8220451837

Median: ma = 21.31990056053
Median: mb = 19.78663589374
Median: mc = 16.90441415044

Inradius: r = 6.27662879989
Circumradius: R = 13.07993157123

Vertex coordinates: A[25; 0] B[0; 0] C[10.82; 16.8220451837]
Centroid: CG[11.94; 5.6076817279]
Coordinates of the circumscribed circle: U[12.5; 3.84994804199]
Coordinates of the inscribed circle: I[11.5; 6.27662879989]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.1321603764° = 130°7'54″ = 0.8770367707 rad
∠ B' = β' = 122.7521739155° = 122°45'6″ = 0.99991706428 rad
∠ C' = γ' = 107.1176657081° = 107°7' = 1.27220543038 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 = 22 ; ; c = 25 ; ;

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

p = a+b+c = 20+22+25 = 67 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67 }{ 2 } = 33.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.5 * (33.5-20)(33.5-22)(33.5-25) } ; ; T = sqrt{ 44207.44 } = 210.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 210.26 }{ 20 } = 21.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 210.26 }{ 22 } = 19.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 210.26 }{ 25 } = 16.82 ; ;

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**2-22**2-25**2 }{ 2 * 22 * 25 } ) = 49° 52'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-20**2-25**2 }{ 2 * 20 * 25 } ) = 57° 14'54" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-20**2-22**2 }{ 2 * 22 * 20 } ) = 72° 53' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 210.26 }{ 33.5 } = 6.28 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 49° 52'6" } = 13.08 ; ;




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