8 10 11 triangle

Acute scalene triangle.

Sides: a = 8   b = 10   c = 11

Area: T = 38.52883986171
Perimeter: p = 29
Semiperimeter: s = 14.5

Angle ∠ A = α = 44.46884446032° = 44°28'6″ = 0.77661207716 rad
Angle ∠ B = β = 61.12114535039° = 61°7'17″ = 1.06767706072 rad
Angle ∠ C = γ = 74.41101018929° = 74°24'36″ = 1.29987012748 rad

Height: ha = 9.63220996543
Height: hb = 7.70656797234
Height: hc = 7.00551633849

Median: ma = 9.72111110476
Median: mb = 8.21658383626
Median: mc = 7.1943747285

Inradius: r = 2.65771309391
Circumradius: R = 5.7110073813

Vertex coordinates: A[11; 0] B[0; 0] C[3.86436363636; 7.00551633849]
Centroid: CG[4.95545454545; 2.33550544616]
Coordinates of the circumscribed circle: U[5.5; 1.53545823372]
Coordinates of the inscribed circle: I[4.5; 2.65771309391]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.5321555397° = 135°31'54″ = 0.77661207716 rad
∠ B' = β' = 118.8798546496° = 118°52'43″ = 1.06767706072 rad
∠ C' = γ' = 105.5989898107° = 105°35'24″ = 1.29987012748 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 = 8 ; ; b = 10 ; ; c = 11 ; ;

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

p = a+b+c = 8+10+11 = 29 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29 }{ 2 } = 14.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.5 * (14.5-8)(14.5-10)(14.5-11) } ; ; T = sqrt{ 1484.44 } = 38.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 38.53 }{ 8 } = 9.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 38.53 }{ 10 } = 7.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 38.53 }{ 11 } = 7.01 ; ;

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{ 8**2-10**2-11**2 }{ 2 * 10 * 11 } ) = 44° 28'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10**2-8**2-11**2 }{ 2 * 8 * 11 } ) = 61° 7'17" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11**2-8**2-10**2 }{ 2 * 10 * 8 } ) = 74° 24'36" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 38.53 }{ 14.5 } = 2.66 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 44° 28'6" } = 5.71 ; ;




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