8 21 22 triangle

Acute scalene triangle.

Sides: a = 8   b = 21   c = 22

Area: T = 83.8365776969
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 21.28799664684° = 21°16'48″ = 0.37114054796 rad
Angle ∠ B = β = 72.30333552301° = 72°18'12″ = 1.2621931609 rad
Angle ∠ C = γ = 86.41766783015° = 86°25' = 1.5088255565 rad

Height: ha = 20.95989442423
Height: hb = 7.98443597113
Height: hc = 7.62114342699

Median: ma = 21.13105466091
Median: mb = 12.79664838921
Median: mc = 11.46773449412

Inradius: r = 3.28876775282
Circumradius: R = 11.02215475231

Vertex coordinates: A[22; 0] B[0; 0] C[2.43218181818; 7.62114342699]
Centroid: CG[8.14439393939; 2.544047809]
Coordinates of the circumscribed circle: U[11; 0.68988467202]
Coordinates of the inscribed circle: I[4.5; 3.28876775282]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.7220033532° = 158°43'12″ = 0.37114054796 rad
∠ B' = β' = 107.697664477° = 107°41'48″ = 1.2621931609 rad
∠ C' = γ' = 93.58333216985° = 93°35' = 1.5088255565 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 = 8 ; ; b = 21 ; ; c = 22 ; ;

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

p = a+b+c = 8+21+22 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-8)(25.5-21)(25.5-22) } ; ; T = sqrt{ 7028.44 } = 83.84 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 83.84 }{ 8 } = 20.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 83.84 }{ 21 } = 7.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 83.84 }{ 22 } = 7.62 ; ;

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{ 8**2-21**2-22**2 }{ 2 * 21 * 22 } ) = 21° 16'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-8**2-22**2 }{ 2 * 8 * 22 } ) = 72° 18'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-8**2-21**2 }{ 2 * 21 * 8 } ) = 86° 25' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 83.84 }{ 25.5 } = 3.29 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 21° 16'48" } = 11.02 ; ;




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