14 21 25 triangle

Acute scalene triangle.

Sides: a = 14   b = 21   c = 25

Area: T = 146.9699384567
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 34.048773237° = 34°2'52″ = 0.59442450327 rad
Angle ∠ B = β = 57.12216504356° = 57°7'18″ = 0.99769608743 rad
Angle ∠ C = γ = 88.83106171944° = 88°49'50″ = 1.55503867466 rad

Height: ha = 20.99656263667
Height: hb = 13.99770842445
Height: hc = 11.75875507654

Median: ma = 22
Median: mb = 17.32877234512
Median: mc = 12.73877392029

Inradius: r = 4.89989794856
Circumradius: R = 12.50326038955

Vertex coordinates: A[25; 0] B[0; 0] C[7.6; 11.75875507654]
Centroid: CG[10.86766666667; 3.91991835885]
Coordinates of the circumscribed circle: U[12.5; 0.25551551815]
Coordinates of the inscribed circle: I[9; 4.89989794856]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.952226763° = 145°57'8″ = 0.59442450327 rad
∠ B' = β' = 122.8788349564° = 122°52'42″ = 0.99769608743 rad
∠ C' = γ' = 91.16993828056° = 91°10'10″ = 1.55503867466 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 = 14 ; ; b = 21 ; ; c = 25 ; ;

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

p = a+b+c = 14+21+25 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-14)(30-21)(30-25) } ; ; T = sqrt{ 21600 } = 146.97 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 146.97 }{ 14 } = 21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 146.97 }{ 21 } = 14 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 146.97 }{ 25 } = 11.76 ; ;

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{ 14**2-21**2-25**2 }{ 2 * 21 * 25 } ) = 34° 2'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-14**2-25**2 }{ 2 * 14 * 25 } ) = 57° 7'18" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-14**2-21**2 }{ 2 * 21 * 14 } ) = 88° 49'50" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 146.97 }{ 30 } = 4.9 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 34° 2'52" } = 12.5 ; ;




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