10 27 29 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 27   c = 29

Area: T = 134.967666255
Perimeter: p = 66
Semiperimeter: s = 33

Angle ∠ A = α = 20.1666073722° = 20°9'58″ = 0.35219643836 rad
Angle ∠ B = β = 68.56105819166° = 68°33'38″ = 1.19766078915 rad
Angle ∠ C = γ = 91.27333443614° = 91°16'24″ = 1.59330203784 rad

Height: ha = 26.99333325101
Height: hb = 9.99875305593
Height: hc = 9.30880456931

Median: ma = 27.56880975042
Median: mb = 16.97879268463
Median: mc = 14.2921605928

Inradius: r = 4.09898988652
Circumradius: R = 14.50435815735

Vertex coordinates: A[29; 0] B[0; 0] C[3.65551724138; 9.30880456931]
Centroid: CG[10.88550574713; 3.10326818977]
Coordinates of the circumscribed circle: U[14.5; -0.32223018127]
Coordinates of the inscribed circle: I[6; 4.09898988652]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.8343926278° = 159°50'2″ = 0.35219643836 rad
∠ B' = β' = 111.4399418083° = 111°26'22″ = 1.19766078915 rad
∠ C' = γ' = 88.72766556386° = 88°43'36″ = 1.59330203784 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 = 27 ; ; c = 29 ; ;

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

p = a+b+c = 10+27+29 = 66 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66 }{ 2 } = 33 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33 * (33-10)(33-27)(33-29) } ; ; T = sqrt{ 18216 } = 134.97 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 134.97 }{ 10 } = 26.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 134.97 }{ 27 } = 10 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 134.97 }{ 29 } = 9.31 ; ;

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**2-27**2-29**2 }{ 2 * 27 * 29 } ) = 20° 9'58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-10**2-29**2 }{ 2 * 10 * 29 } ) = 68° 33'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-10**2-27**2 }{ 2 * 27 * 10 } ) = 91° 16'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 134.97 }{ 33 } = 4.09 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 20° 9'58" } = 14.5 ; ;




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