20 22 30 triangle

Obtuse scalene triangle.

Sides: a = 20   b = 22   c = 30

Area: T = 219.9643633358
Perimeter: p = 72
Semiperimeter: s = 36

Angle ∠ A = α = 41.80218441931° = 41°48'7″ = 0.73295798146 rad
Angle ∠ B = β = 47.15663569564° = 47°9'23″ = 0.82330336921 rad
Angle ∠ C = γ = 91.04217988505° = 91°2'30″ = 1.58989791469 rad

Height: ha = 21.99663633358
Height: hb = 19.99766939416
Height: hc = 14.66442422239

Median: ma = 24.33110501212
Median: mb = 23
Median: mc = 14.73109198627

Inradius: r = 6.11101009266
Circumradius: R = 15.00224799537

Vertex coordinates: A[30; 0] B[0; 0] C[13.6; 14.66442422239]
Centroid: CG[14.53333333333; 4.88880807413]
Coordinates of the circumscribed circle: U[15; -0.27327723628]
Coordinates of the inscribed circle: I[14; 6.11101009266]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.1988155807° = 138°11'53″ = 0.73295798146 rad
∠ B' = β' = 132.8443643044° = 132°50'37″ = 0.82330336921 rad
∠ C' = γ' = 88.95882011495° = 88°57'30″ = 1.58989791469 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 = 20 ; ; b = 22 ; ; c = 30 ; ;

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

p = a+b+c = 20+22+30 = 72 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72 }{ 2 } = 36 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36 * (36-20)(36-22)(36-30) } ; ; T = sqrt{ 48384 } = 219.96 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 219.96 }{ 20 } = 22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 219.96 }{ 22 } = 20 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 219.96 }{ 30 } = 14.66 ; ;

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{ 20**2-22**2-30**2 }{ 2 * 22 * 30 } ) = 41° 48'7" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-20**2-30**2 }{ 2 * 20 * 30 } ) = 47° 9'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-20**2-22**2 }{ 2 * 22 * 20 } ) = 91° 2'30" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 219.96 }{ 36 } = 6.11 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 41° 48'7" } = 15 ; ;




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