21 22 27 triangle

Acute scalene triangle.

Sides: a = 21   b = 22   c = 27

Area: T = 225.7433216952
Perimeter: p = 70
Semiperimeter: s = 35

Angle ∠ A = α = 49.47110898048° = 49°28'16″ = 0.86334334016 rad
Angle ∠ B = β = 52.77655999225° = 52°46'32″ = 0.92111079834 rad
Angle ∠ C = γ = 77.75333102727° = 77°45'12″ = 1.35770512686 rad

Height: ha = 21.49993539955
Height: hb = 20.5222110632
Height: hc = 16.72217197742

Median: ma = 22.27766694099
Median: mb = 21.54106592285
Median: mc = 16.74106690428

Inradius: r = 6.45498061986
Circumradius: R = 13.81443685649

Vertex coordinates: A[27; 0] B[0; 0] C[12.70437037037; 16.72217197742]
Centroid: CG[13.23545679012; 5.57439065914]
Coordinates of the circumscribed circle: U[13.5; 2.93303206047]
Coordinates of the inscribed circle: I[13; 6.45498061986]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.5298910195° = 130°31'44″ = 0.86334334016 rad
∠ B' = β' = 127.2244400078° = 127°13'28″ = 0.92111079834 rad
∠ C' = γ' = 102.2476689727° = 102°14'48″ = 1.35770512686 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 = 22 ; ; c = 27 ; ;

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

p = a+b+c = 21+22+27 = 70 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 70 }{ 2 } = 35 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35 * (35-21)(35-22)(35-27) } ; ; T = sqrt{ 50960 } = 225.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 225.74 }{ 21 } = 21.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 225.74 }{ 22 } = 20.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 225.74 }{ 27 } = 16.72 ; ;

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**2-22**2-27**2 }{ 2 * 22 * 27 } ) = 49° 28'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 52° 46'32" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-21**2-22**2 }{ 2 * 22 * 21 } ) = 77° 45'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 225.74 }{ 35 } = 6.45 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 49° 28'16" } = 13.81 ; ;




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