15 21 22 triangle

Acute scalene triangle.

Sides: a = 15   b = 21   c = 22

Area: T = 150.78546146
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 40.74990543775° = 40°44'57″ = 0.7111205166 rad
Angle ∠ B = β = 66.04223927371° = 66°2'33″ = 1.15326571992 rad
Angle ∠ C = γ = 73.20985528854° = 73°12'31″ = 1.27877302885 rad

Height: ha = 20.105461528
Height: hb = 14.36604394857
Height: hc = 13.70876922363

Median: ma = 20.15656443707
Median: mb = 15.62884996081
Median: mc = 14.56602197786

Inradius: r = 5.1999469469
Circumradius: R = 11.49898990497

Vertex coordinates: A[22; 0] B[0; 0] C[6.09109090909; 13.70876922363]
Centroid: CG[9.36436363636; 4.56992307454]
Coordinates of the circumscribed circle: U[11; 3.31993041699]
Coordinates of the inscribed circle: I[8; 5.1999469469]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.2510945623° = 139°15'3″ = 0.7111205166 rad
∠ B' = β' = 113.9587607263° = 113°57'27″ = 1.15326571992 rad
∠ C' = γ' = 106.7911447115° = 106°47'29″ = 1.27877302885 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 = 15 ; ; b = 21 ; ; c = 22 ; ;

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

p = a+b+c = 15+21+22 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-15)(29-21)(29-22) } ; ; T = sqrt{ 22736 } = 150.78 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 150.78 }{ 15 } = 20.1 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 150.78 }{ 21 } = 14.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 150.78 }{ 22 } = 13.71 ; ;

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{ 15**2-21**2-22**2 }{ 2 * 21 * 22 } ) = 40° 44'57" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-15**2-22**2 }{ 2 * 15 * 22 } ) = 66° 2'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-15**2-21**2 }{ 2 * 21 * 15 } ) = 73° 12'31" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 150.78 }{ 29 } = 5.2 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 40° 44'57" } = 11.49 ; ;




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