6 8 9 triangle

Acute scalene triangle.

Sides: a = 6   b = 8   c = 9

Area: T = 23.52552523897
Perimeter: p = 23
Semiperimeter: s = 11.5

Angle ∠ A = α = 40.80444376906° = 40°48'16″ = 0.71221717871 rad
Angle ∠ B = β = 60.61107200521° = 60°36'39″ = 1.05878566269 rad
Angle ∠ C = γ = 78.58548422573° = 78°35'5″ = 1.37215642395 rad

Height: ha = 7.84217507966
Height: hb = 5.88113130974
Height: hc = 5.22878338644

Median: ma = 7.96986887253
Median: mb = 6.51992024052
Median: mc = 5.45443560573

Inradius: r = 2.04656741208
Circumradius: R = 4.59108115335

Vertex coordinates: A[9; 0] B[0; 0] C[2.94444444444; 5.22878338644]
Centroid: CG[3.98114814815; 1.74326112881]
Coordinates of the circumscribed circle: U[4.5; 0.9098598116]
Coordinates of the inscribed circle: I[3.5; 2.04656741208]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.1965562309° = 139°11'44″ = 0.71221717871 rad
∠ B' = β' = 119.3899279948° = 119°23'21″ = 1.05878566269 rad
∠ C' = γ' = 101.4155157743° = 101°24'55″ = 1.37215642395 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 = 6 ; ; b = 8 ; ; c = 9 ; ;

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

p = a+b+c = 6+8+9 = 23 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23 }{ 2 } = 11.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.5 * (11.5-6)(11.5-8)(11.5-9) } ; ; T = sqrt{ 553.44 } = 23.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 23.53 }{ 6 } = 7.84 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 23.53 }{ 8 } = 5.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 23.53 }{ 9 } = 5.23 ; ;

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{ 6**2-8**2-9**2 }{ 2 * 8 * 9 } ) = 40° 48'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8**2-6**2-9**2 }{ 2 * 6 * 9 } ) = 60° 36'39" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9**2-6**2-8**2 }{ 2 * 8 * 6 } ) = 78° 35'5" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 23.53 }{ 11.5 } = 2.05 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6 }{ 2 * sin 40° 48'16" } = 4.59 ; ;




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