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: h_a = 20.72770204082
Height: h_b = 3.94880038873
Height: h_c = 3.76985491651

Median: m_a = 21.41326131054
Median: m_b = 11.82215904175
Median: m_c = 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

Calculate another triangle

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

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

6. Inradius

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

7. Circumradius

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

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