21 25 30 triangle

Acute scalene triangle.

Sides: a = 21   b = 25   c = 30

Area: T = 259.1998765429
Perimeter: p = 76
Semiperimeter: s = 38

Angle ∠ A = α = 43.72549134683° = 43°43'30″ = 0.76331437052 rad
Angle ∠ B = β = 55.37114408187° = 55°22'17″ = 0.96664139539 rad
Angle ∠ C = γ = 80.9043645713° = 80°54'13″ = 1.41220349946 rad

Height: ha = 24.68655967075
Height: hb = 20.73659012343
Height: hc = 17.28799176953

Median: ma = 25.53991855782
Median: mb = 22.67770809409
Median: mc = 17.55499287748

Inradius: r = 6.82110201429
Circumradius: R = 15.19110445772

Vertex coordinates: A[30; 0] B[0; 0] C[11.93333333333; 17.28799176953]
Centroid: CG[13.97877777778; 5.76599725651]
Coordinates of the circumscribed circle: U[15; 2.40216318094]
Coordinates of the inscribed circle: I[13; 6.82110201429]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.2755086532° = 136°16'30″ = 0.76331437052 rad
∠ B' = β' = 124.6298559181° = 124°37'43″ = 0.96664139539 rad
∠ C' = γ' = 99.0966354287° = 99°5'47″ = 1.41220349946 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 = 30 ; ;

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

p = a+b+c = 21+25+30 = 76 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 76 }{ 2 } = 38 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38 * (38-21)(38-25)(38-30) } ; ; T = sqrt{ 67184 } = 259.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 259.2 }{ 21 } = 24.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 259.2 }{ 25 } = 20.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 259.2 }{ 30 } = 17.28 ; ;

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**2-30**2 }{ 2 * 25 * 30 } ) = 43° 43'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 25**2-21**2-30**2 }{ 2 * 21 * 30 } ) = 55° 22'17" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-21**2-25**2 }{ 2 * 25 * 21 } ) = 80° 54'13" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 259.2 }{ 38 } = 6.82 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 43° 43'30" } = 15.19 ; ;




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