21 29 30 triangle

Acute scalene triangle.

Sides: a = 21   b = 29   c = 30

Area: T = 289.1376645896
Perimeter: p = 80
Semiperimeter: s = 40

Angle ∠ A = α = 41.65879296759° = 41°39'29″ = 0.72770680324 rad
Angle ∠ B = β = 66.62201318843° = 66°37'12″ = 1.16327406495 rad
Angle ∠ C = γ = 71.72219384398° = 71°43'19″ = 1.25217839717 rad

Height: ha = 27.53768234187
Height: hb = 19.94404583377
Height: hc = 19.27657763931

Median: ma = 27.57326313579
Median: mb = 21.45334379529
Median: mc = 20.39660780544

Inradius: r = 7.22884161474
Circumradius: R = 15.79770290685

Vertex coordinates: A[30; 0] B[0; 0] C[8.33333333333; 19.27657763931]
Centroid: CG[12.77877777778; 6.42552587977]
Coordinates of the circumscribed circle: U[15; 4.95444048474]
Coordinates of the inscribed circle: I[11; 7.22884161474]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.3422070324° = 138°20'31″ = 0.72770680324 rad
∠ B' = β' = 113.3879868116° = 113°22'48″ = 1.16327406495 rad
∠ C' = γ' = 108.278806156° = 108°16'41″ = 1.25217839717 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 = 29 ; ; c = 30 ; ;

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

p = a+b+c = 21+29+30 = 80 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 80 }{ 2 } = 40 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 40 * (40-21)(40-29)(40-30) } ; ; T = sqrt{ 83600 } = 289.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 289.14 }{ 21 } = 27.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 289.14 }{ 29 } = 19.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 289.14 }{ 30 } = 19.28 ; ;

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**2-29**2-30**2 }{ 2 * 29 * 30 } ) = 41° 39'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 29**2-21**2-30**2 }{ 2 * 21 * 30 } ) = 66° 37'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-21**2-29**2 }{ 2 * 29 * 21 } ) = 71° 43'19" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 289.14 }{ 40 } = 7.23 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 41° 39'29" } = 15.8 ; ;




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