11 23 24 triangle

Acute scalene triangle.

Sides: a = 11   b = 23   c = 24

Area: T = 125.1439921688
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 26.96223889975° = 26°57'45″ = 0.47105824622 rad
Angle ∠ B = β = 71.4476995465° = 71°26'49″ = 1.24769853115 rad
Angle ∠ C = γ = 81.59106155375° = 81°35'26″ = 1.42440248799 rad

Height: ha = 22.75327130341
Height: hb = 10.88217323207
Height: hc = 10.42883268073

Median: ma = 22.85327897641
Median: mb = 14.70554411699
Median: mc = 13.45436240471

Inradius: r = 4.31551697134
Circumradius: R = 12.13304215276

Vertex coordinates: A[24; 0] B[0; 0] C[3.5; 10.42883268073]
Centroid: CG[9.16766666667; 3.47661089358]
Coordinates of the circumscribed circle: U[12; 1.77440142155]
Coordinates of the inscribed circle: I[6; 4.31551697134]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.0387611003° = 153°2'15″ = 0.47105824622 rad
∠ B' = β' = 108.5533004535° = 108°33'11″ = 1.24769853115 rad
∠ C' = γ' = 98.40993844625° = 98°24'34″ = 1.42440248799 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 = 23 ; ; c = 24 ; ;

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

p = a+b+c = 11+23+24 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-11)(29-23)(29-24) } ; ; T = sqrt{ 15660 } = 125.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 125.14 }{ 11 } = 22.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 125.14 }{ 23 } = 10.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 125.14 }{ 24 } = 10.43 ; ;

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**2-23**2-24**2 }{ 2 * 23 * 24 } ) = 26° 57'45" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-11**2-24**2 }{ 2 * 11 * 24 } ) = 71° 26'49" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-11**2-23**2 }{ 2 * 23 * 11 } ) = 81° 35'26" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 125.14 }{ 29 } = 4.32 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 26° 57'45" } = 12.13 ; ;




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