6 11 12 triangle

Acute scalene triangle.

Sides: a = 6   b = 11   c = 12

Area: T = 32.84395721653
Perimeter: p = 29
Semiperimeter: s = 14.5

Angle ∠ A = α = 29.83993144221° = 29°50'22″ = 0.52107942832 rad
Angle ∠ B = β = 65.81326135681° = 65°48'45″ = 1.14986467961 rad
Angle ∠ C = γ = 84.34880720098° = 84°20'53″ = 1.47221515743 rad

Height: ha = 10.94765240551
Height: hb = 5.97108313028
Height: hc = 5.47332620276

Median: ma = 11.11330553854
Median: mb = 7.73298124169
Median: mc = 6.51992024052

Inradius: r = 2.26547980804
Circumradius: R = 6.02993111921

Vertex coordinates: A[12; 0] B[0; 0] C[2.45883333333; 5.47332620276]
Centroid: CG[4.81994444444; 1.82444206759]
Coordinates of the circumscribed circle: U[6; 0.59437957992]
Coordinates of the inscribed circle: I[3.5; 2.26547980804]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.1610685578° = 150°9'38″ = 0.52107942832 rad
∠ B' = β' = 114.1877386432° = 114°11'15″ = 1.14986467961 rad
∠ C' = γ' = 95.65219279902° = 95°39'7″ = 1.47221515743 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 = 11 ; ; c = 12 ; ;

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

p = a+b+c = 6+11+12 = 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 * (14.5-6)(14.5-11)(14.5-12) } ; ; T = sqrt{ 1078.44 } = 32.84 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 32.84 }{ 6 } = 10.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 32.84 }{ 11 } = 5.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 32.84 }{ 12 } = 5.47 ; ;

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**2-11**2-12**2 }{ 2 * 11 * 12 } ) = 29° 50'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11**2-6**2-12**2 }{ 2 * 6 * 12 } ) = 65° 48'45" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12**2-6**2-11**2 }{ 2 * 11 * 6 } ) = 84° 20'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 32.84 }{ 14.5 } = 2.26 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6 }{ 2 * sin 29° 50'22" } = 6.03 ; ;




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