9 11 15 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 11   c = 15

Area: T = 49.16549010982
Perimeter: p = 35
Semiperimeter: s = 17.5

Angle ∠ A = α = 36.58795428375° = 36°34'46″ = 0.63884334614 rad
Angle ∠ B = β = 46.75498273458° = 46°44'59″ = 0.81659384119 rad
Angle ∠ C = γ = 96.67106298167° = 96°40'14″ = 1.68772207803 rad

Height: ha = 10.92655335774
Height: hb = 8.9399072927
Height: hc = 6.55553201464

Median: ma = 12.35992070943
Median: mb = 11.07992599031
Median: mc = 6.69895440801

Inradius: r = 2.80994229199
Circumradius: R = 7.55111186173

Vertex coordinates: A[15; 0] B[0; 0] C[6.16766666667; 6.55553201464]
Centroid: CG[7.05655555556; 2.18551067155]
Coordinates of the circumscribed circle: U[7.5; -0.87771501424]
Coordinates of the inscribed circle: I[6.5; 2.80994229199]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.4220457162° = 143°25'14″ = 0.63884334614 rad
∠ B' = β' = 133.2550172654° = 133°15'1″ = 0.81659384119 rad
∠ C' = γ' = 83.32993701833° = 83°19'46″ = 1.68772207803 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 = 11 ; ; c = 15 ; ;

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

p = a+b+c = 9+11+15 = 35 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 35 }{ 2 } = 17.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.5 * (17.5-9)(17.5-11)(17.5-15) } ; ; T = sqrt{ 2417.19 } = 49.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 49.16 }{ 9 } = 10.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 49.16 }{ 11 } = 8.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 49.16 }{ 15 } = 6.56 ; ;

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**2-11**2-15**2 }{ 2 * 11 * 15 } ) = 36° 34'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11**2-9**2-15**2 }{ 2 * 9 * 15 } ) = 46° 44'59" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-9**2-11**2 }{ 2 * 11 * 9 } ) = 96° 40'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 49.16 }{ 17.5 } = 2.81 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 36° 34'46" } = 7.55 ; ;




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