11 12 15 triangle

Acute scalene triangle.

Sides: a = 11   b = 12   c = 15

Area: T = 65.23880257212
Perimeter: p = 38
Semiperimeter: s = 19

Angle ∠ A = α = 46.45877809718° = 46°27'28″ = 0.81108412411 rad
Angle ∠ B = β = 52.25769610468° = 52°15'25″ = 0.91220560274 rad
Angle ∠ C = γ = 81.28552579814° = 81°17'7″ = 1.41986953851 rad

Height: ha = 11.8611459222
Height: hb = 10.87330042869
Height: hc = 8.69884034295

Median: ma = 12.42197423484
Median: mb = 11.70546999107
Median: mc = 8.73221245983

Inradius: r = 3.43435803011
Circumradius: R = 7.58875993261

Vertex coordinates: A[15; 0] B[0; 0] C[6.73333333333; 8.69884034295]
Centroid: CG[7.24444444444; 2.89994678098]
Coordinates of the circumscribed circle: U[7.5; 1.15496362615]
Coordinates of the inscribed circle: I[7; 3.43435803011]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.5422219028° = 133°32'32″ = 0.81108412411 rad
∠ B' = β' = 127.7433038953° = 127°44'35″ = 0.91220560274 rad
∠ C' = γ' = 98.71547420186° = 98°42'53″ = 1.41986953851 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 = 11 ; ; b = 12 ; ; c = 15 ; ;

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

p = a+b+c = 11+12+15 = 38 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38 }{ 2 } = 19 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19 * (19-11)(19-12)(19-15) } ; ; T = sqrt{ 4256 } = 65.24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 65.24 }{ 11 } = 11.86 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 65.24 }{ 12 } = 10.87 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 65.24 }{ 15 } = 8.7 ; ;

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{ 11**2-12**2-15**2 }{ 2 * 12 * 15 } ) = 46° 27'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-11**2-15**2 }{ 2 * 11 * 15 } ) = 52° 15'25" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-11**2-12**2 }{ 2 * 12 * 11 } ) = 81° 17'7" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 65.24 }{ 19 } = 3.43 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 46° 27'28" } = 7.59 ; ;




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