4 21 22 triangle

Obtuse scalene triangle.

Sides: a = 4   b = 21   c = 22

Area: T = 41.45440408163
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 10.33880005321° = 10°20'17″ = 0.18804321474 rad
Angle ∠ B = β = 70.414364113° = 70°24'49″ = 1.2298949876 rad
Angle ∠ C = γ = 99.24883583379° = 99°14'54″ = 1.73222106302 rad

Height: ha = 20.72770204082
Height: hb = 3.94880038873
Height: hc = 3.76985491651

Median: ma = 21.41326131054
Median: mb = 11.82215904175
Median: mc = 10.36882206767

Inradius: r = 1.76440017369
Circumradius: R = 11.14548725119

Vertex coordinates: A[22; 0] B[0; 0] C[1.34109090909; 3.76985491651]
Centroid: CG[7.78803030303; 1.2566183055]
Coordinates of the circumscribed circle: U[11; -1.79111402251]
Coordinates of the inscribed circle: I[2.5; 1.76440017369]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 169.6621999468° = 169°39'43″ = 0.18804321474 rad
∠ B' = β' = 109.586635887° = 109°35'11″ = 1.2298949876 rad
∠ C' = γ' = 80.75216416621° = 80°45'6″ = 1.73222106302 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 = 4 ; ; b = 21 ; ; c = 22 ; ;

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

p = a+b+c = 4+21+22 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-4)(23.5-21)(23.5-22) } ; ; T = sqrt{ 1718.44 } = 41.45 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 41.45 }{ 4 } = 20.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 41.45 }{ 21 } = 3.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 41.45 }{ 22 } = 3.77 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 21**2+22**2-4**2 }{ 2 * 21 * 22 } ) = 10° 20'17" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4**2+22**2-21**2 }{ 2 * 4 * 22 } ) = 70° 24'49" ; ; gamma = 180° - alpha - beta = 180° - 10° 20'17" - 70° 24'49" = 99° 14'54" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 41.45 }{ 23.5 } = 1.76 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4 }{ 2 * sin 10° 20'17" } = 11.14 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 22**2 - 4**2 } }{ 2 } = 21.413 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 22**2+2 * 4**2 - 21**2 } }{ 2 } = 11.822 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 4**2 - 22**2 } }{ 2 } = 10.368 ; ;
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