21 25 28 triangle

Acute scalene triangle.

Sides: a = 21   b = 25   c = 28

Area: T = 252.8565690068
Perimeter: p = 74
Semiperimeter: s = 37

Angle ∠ A = α = 46.25767006748° = 46°15'24″ = 0.80773317279 rad
Angle ∠ B = β = 59.32325755274° = 59°19'21″ = 1.03553742637 rad
Angle ∠ C = γ = 74.42107237978° = 74°25'15″ = 1.2998886662 rad

Height: ha = 24.08114942922
Height: hb = 20.22884552055
Height: hc = 18.06111207192

Median: ma = 24.37772434865
Median: mb = 21.36600093633
Median: mc = 18.35875597507

Inradius: r = 6.83439375694
Circumradius: R = 14.534398181

Vertex coordinates: A[28; 0] B[0; 0] C[10.71442857143; 18.06111207192]
Centroid: CG[12.90547619048; 6.02203735731]
Coordinates of the circumscribed circle: U[14; 3.90334122575]
Coordinates of the inscribed circle: I[12; 6.83439375694]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.7433299325° = 133°44'36″ = 0.80773317279 rad
∠ B' = β' = 120.6777424473° = 120°40'39″ = 1.03553742637 rad
∠ C' = γ' = 105.5799276202° = 105°34'45″ = 1.2998886662 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 = 21 ; ; b = 25 ; ; c = 28 ; ;

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

p = a+b+c = 21+25+28 = 74 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 74 }{ 2 } = 37 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37 * (37-21)(37-25)(37-28) } ; ; T = sqrt{ 63936 } = 252.86 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 252.86 }{ 21 } = 24.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 252.86 }{ 25 } = 20.23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 252.86 }{ 28 } = 18.06 ; ;

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{ 21**2-25**2-28**2 }{ 2 * 25 * 28 } ) = 46° 15'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 25**2-21**2-28**2 }{ 2 * 21 * 28 } ) = 59° 19'21" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-21**2-25**2 }{ 2 * 25 * 21 } ) = 74° 25'15" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 252.86 }{ 37 } = 6.83 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 46° 15'24" } = 14.53 ; ;




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