18 21 22 triangle

Acute scalene triangle.

Sides: a = 18   b = 21   c = 22

Area: T = 175.4599218909
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 49.42657513336° = 49°25'33″ = 0.8632642096 rad
Angle ∠ B = β = 62.39443231026° = 62°23'40″ = 1.08989863727 rad
Angle ∠ C = γ = 68.18799255638° = 68°10'48″ = 1.19899641849 rad

Height: ha = 19.49554687676
Height: hb = 16.71104018008
Height: hc = 15.95108380826

Median: ma = 19.53220249846
Median: mb = 17.1399136501
Median: mc = 16.17109616288

Inradius: r = 5.75327612757
Circumradius: R = 11.84989071873

Vertex coordinates: A[22; 0] B[0; 0] C[8.34109090909; 15.95108380826]
Centroid: CG[10.11436363636; 5.31769460275]
Coordinates of the circumscribed circle: U[11; 4.40441573011]
Coordinates of the inscribed circle: I[9.5; 5.75327612757]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.5744248666° = 130°34'27″ = 0.8632642096 rad
∠ B' = β' = 117.6065676897° = 117°36'20″ = 1.08989863727 rad
∠ C' = γ' = 111.8220074436° = 111°49'12″ = 1.19899641849 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 = 18 ; ; b = 21 ; ; c = 22 ; ;

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

p = a+b+c = 18+21+22 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-18)(30.5-21)(30.5-22) } ; ; T = sqrt{ 30785.94 } = 175.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 175.46 }{ 18 } = 19.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 175.46 }{ 21 } = 16.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 175.46 }{ 22 } = 15.95 ; ;

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{ 18**2-21**2-22**2 }{ 2 * 21 * 22 } ) = 49° 25'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-18**2-22**2 }{ 2 * 18 * 22 } ) = 62° 23'40" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-18**2-21**2 }{ 2 * 21 * 18 } ) = 68° 10'48" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 175.46 }{ 30.5 } = 5.75 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 49° 25'33" } = 11.85 ; ;




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