11 21 24 triangle

Obtuse scalene triangle.

Sides: a = 11   b = 21   c = 24

Area: T = 115.4476957517
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 27.26660444507° = 27°15'58″ = 0.47658822497 rad
Angle ∠ B = β = 60.99774539733° = 60°59'51″ = 1.06546064072 rad
Angle ∠ C = γ = 91.73765015759° = 91°44'11″ = 1.60111039968 rad

Height: ha = 20.99903559122
Height: hb = 10.9954948335
Height: hc = 9.62105797931

Median: ma = 21.86989277286
Median: mb = 15.43553490404
Median: mc = 11.70546999107

Inradius: r = 4.12331056256
Circumradius: R = 12.00655134393

Vertex coordinates: A[24; 0] B[0; 0] C[5.33333333333; 9.62105797931]
Centroid: CG[9.77877777778; 3.2076859931]
Coordinates of the circumscribed circle: U[12; -0.36438034376]
Coordinates of the inscribed circle: I[7; 4.12331056256]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.7343955549° = 152°44'2″ = 0.47658822497 rad
∠ B' = β' = 119.0032546027° = 119°9″ = 1.06546064072 rad
∠ C' = γ' = 88.26334984241° = 88°15'49″ = 1.60111039968 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 = 21 ; ; c = 24 ; ;

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

p = a+b+c = 11+21+24 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-11)(28-21)(28-24) } ; ; T = sqrt{ 13328 } = 115.45 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 115.45 }{ 11 } = 20.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 115.45 }{ 21 } = 10.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 115.45 }{ 24 } = 9.62 ; ;

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**2-21**2-24**2 }{ 2 * 21 * 24 } ) = 27° 15'58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-11**2-24**2 }{ 2 * 11 * 24 } ) = 60° 59'51" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-11**2-21**2 }{ 2 * 21 * 11 } ) = 91° 44'11" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 115.45 }{ 28 } = 4.12 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 27° 15'58" } = 12.01 ; ;




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