10 22 30 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 22   c = 30

Area: T = 76.5444104933
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 13.41220116047° = 13°24'43″ = 0.23440837618 rad
Angle ∠ B = β = 30.6833417109° = 30°41' = 0.53655266543 rad
Angle ∠ C = γ = 135.9054571286° = 135°54'16″ = 2.37219822375 rad

Height: ha = 15.30988209866
Height: hb = 6.95985549939
Height: hc = 5.10329403289

Median: ma = 25.82663431403
Median: mb = 19.46879223339
Median: mc = 8.18553527719

Inradius: r = 2.46991646753
Circumradius: R = 21.55661995459

Vertex coordinates: A[30; 0] B[0; 0] C[8.6; 5.10329403289]
Centroid: CG[12.86766666667; 1.70109801096]
Coordinates of the circumscribed circle: U[15; -15.4811270583]
Coordinates of the inscribed circle: I[9; 2.46991646753]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 166.5887988395° = 166°35'17″ = 0.23440837618 rad
∠ B' = β' = 149.3176582891° = 149°19' = 0.53655266543 rad
∠ C' = γ' = 44.09554287137° = 44°5'44″ = 2.37219822375 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 = 10 ; ; b = 22 ; ; c = 30 ; ;

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

p = a+b+c = 10+22+30 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-10)(31-22)(31-30) } ; ; T = sqrt{ 5859 } = 76.54 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 76.54 }{ 10 } = 15.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 76.54 }{ 22 } = 6.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 76.54 }{ 30 } = 5.1 ; ;

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{ 10**2-22**2-30**2 }{ 2 * 22 * 30 } ) = 13° 24'43" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-10**2-30**2 }{ 2 * 10 * 30 } ) = 30° 41' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-10**2-22**2 }{ 2 * 22 * 10 } ) = 135° 54'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 76.54 }{ 31 } = 2.47 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 13° 24'43" } = 21.56 ; ;




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