9 10 11 triangle

Acute scalene triangle.

Sides: a = 9   b = 10   c = 11

Area: T = 42.42664068712
Perimeter: p = 30
Semiperimeter: s = 15

Angle ∠ A = α = 50.47988036414° = 50°28'44″ = 0.8811021326 rad
Angle ∠ B = β = 58.99224169931° = 58°59'33″ = 1.03296119102 rad
Angle ∠ C = γ = 70.52987793655° = 70°31'44″ = 1.23109594173 rad

Height: ha = 9.42880904158
Height: hb = 8.48552813742
Height: hc = 7.71438921584

Median: ma = 9.5
Median: mb = 8.71877978871
Median: mc = 7.76220873481

Inradius: r = 2.82884271247
Circumradius: R = 5.83436309448

Vertex coordinates: A[11; 0] B[0; 0] C[4.63663636364; 7.71438921584]
Centroid: CG[5.21221212121; 2.57112973861]
Coordinates of the circumscribed circle: U[5.5; 1.94545436483]
Coordinates of the inscribed circle: I[5; 2.82884271247]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.5211196359° = 129°31'16″ = 0.8811021326 rad
∠ B' = β' = 121.0087583007° = 121°27″ = 1.03296119102 rad
∠ C' = γ' = 109.4711220634° = 109°28'16″ = 1.23109594173 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 = 9 ; ; b = 10 ; ; c = 11 ; ;

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

p = a+b+c = 9+10+11 = 30 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30 }{ 2 } = 15 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15 * (15-9)(15-10)(15-11) } ; ; T = sqrt{ 1800 } = 42.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 42.43 }{ 9 } = 9.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 42.43 }{ 10 } = 8.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 42.43 }{ 11 } = 7.71 ; ;

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{ 9**2-10**2-11**2 }{ 2 * 10 * 11 } ) = 50° 28'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10**2-9**2-11**2 }{ 2 * 9 * 11 } ) = 58° 59'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11**2-9**2-10**2 }{ 2 * 10 * 9 } ) = 70° 31'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 42.43 }{ 15 } = 2.83 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 50° 28'44" } = 5.83 ; ;




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