11 13 23 triangle

Obtuse scalene triangle.

Sides: a = 11   b = 13   c = 23

Area: T = 39.2710695181
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 15.22991574517° = 15°13'45″ = 0.26657989398 rad
Angle ∠ B = β = 18.08657656893° = 18°5'9″ = 0.3165656159 rad
Angle ∠ C = γ = 146.6855076859° = 146°41'6″ = 2.56601375547 rad

Height: ha = 7.14401263965
Height: hb = 6.04216454125
Height: hc = 3.41548430592

Median: ma = 17.85435710714
Median: mb = 16.81551717208
Median: mc = 3.57107142143

Inradius: r = 1.6711093412
Circumradius: R = 20.93880046931

Vertex coordinates: A[23; 0] B[0; 0] C[10.45765217391; 3.41548430592]
Centroid: CG[11.1522173913; 1.13882810197]
Coordinates of the circumscribed circle: U[11.5; -17.4977143782]
Coordinates of the inscribed circle: I[10.5; 1.6711093412]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 164.7710842548° = 164°46'15″ = 0.26657989398 rad
∠ B' = β' = 161.9144234311° = 161°54'51″ = 0.3165656159 rad
∠ C' = γ' = 33.31549231411° = 33°18'54″ = 2.56601375547 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 = 13 ; ; c = 23 ; ;

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

p = a+b+c = 11+13+23 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-11)(23.5-13)(23.5-23) } ; ; T = sqrt{ 1542.19 } = 39.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 39.27 }{ 11 } = 7.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 39.27 }{ 13 } = 6.04 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 39.27 }{ 23 } = 3.41 ; ;

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**2-13**2-23**2 }{ 2 * 13 * 23 } ) = 15° 13'45" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-11**2-23**2 }{ 2 * 11 * 23 } ) = 18° 5'9" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-11**2-13**2 }{ 2 * 13 * 11 } ) = 146° 41'6" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 39.27 }{ 23.5 } = 1.67 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 15° 13'45" } = 20.94 ; ;




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