7 21 22 triangle

Acute scalene triangle.

Sides: a = 7   b = 21   c = 22

Area: T = 73.48546922835
Perimeter: p = 50
Semiperimeter: s = 25

Angle ∠ A = α = 18.54989996134° = 18°32'56″ = 0.32437411162 rad
Angle ∠ B = β = 72.62203831922° = 72°37'13″ = 1.26774647908 rad
Angle ∠ C = γ = 88.83106171944° = 88°49'50″ = 1.55503867466 rad

Height: ha = 20.99656263667
Height: hb = 6.99985421222
Height: hc = 6.68804265712

Median: ma = 21.21990951739
Median: mb = 12.5
Median: mc = 11.13655287257

Inradius: r = 2.93993876913
Circumradius: R = 11.0022291428

Vertex coordinates: A[22; 0] B[0; 0] C[2.09109090909; 6.68804265712]
Centroid: CG[8.03303030303; 2.22768088571]
Coordinates of the circumscribed circle: U[11; 0.22545365598]
Coordinates of the inscribed circle: I[4; 2.93993876913]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.4511000387° = 161°27'4″ = 0.32437411162 rad
∠ B' = β' = 107.3879616808° = 107°22'47″ = 1.26774647908 rad
∠ C' = γ' = 91.16993828056° = 91°10'10″ = 1.55503867466 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 = 7 ; ; b = 21 ; ; c = 22 ; ;

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

p = a+b+c = 7+21+22 = 50 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 50 }{ 2 } = 25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25 * (25-7)(25-21)(25-22) } ; ; T = sqrt{ 5400 } = 73.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 73.48 }{ 7 } = 21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 73.48 }{ 21 } = 7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 73.48 }{ 22 } = 6.68 ; ;

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{ 7**2-21**2-22**2 }{ 2 * 21 * 22 } ) = 18° 32'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-7**2-22**2 }{ 2 * 7 * 22 } ) = 72° 37'13" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-7**2-21**2 }{ 2 * 21 * 7 } ) = 88° 49'50" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 73.48 }{ 25 } = 2.94 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7 }{ 2 * sin 18° 32'56" } = 11 ; ;




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