14 21 24 triangle

Acute scalene triangle.

Sides: a = 14   b = 21   c = 24

Area: T = 146.2076831236
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 35.46436145718° = 35°27'49″ = 0.61989568389 rad
Angle ∠ B = β = 60.49110490724° = 60°29'28″ = 1.05657679743 rad
Angle ∠ C = γ = 84.04553363559° = 84°2'43″ = 1.46768678404 rad

Height: ha = 20.88766901765
Height: hb = 13.92444601177
Height: hc = 12.1843902603

Median: ma = 21.43659511102
Median: mb = 16.60657219054
Median: mc = 13.21098448136

Inradius: r = 4.95661637707
Circumradius: R = 12.06550997295

Vertex coordinates: A[24; 0] B[0; 0] C[6.89658333333; 12.1843902603]
Centroid: CG[10.29986111111; 4.06113008677]
Coordinates of the circumscribed circle: U[12; 1.25216515026]
Coordinates of the inscribed circle: I[8.5; 4.95661637707]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.5366385428° = 144°32'11″ = 0.61989568389 rad
∠ B' = β' = 119.5098950928° = 119°30'32″ = 1.05657679743 rad
∠ C' = γ' = 95.95546636441° = 95°57'17″ = 1.46768678404 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 = 14 ; ; b = 21 ; ; c = 24 ; ;

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

p = a+b+c = 14+21+24 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-14)(29.5-21)(29.5-24) } ; ; T = sqrt{ 21376.44 } = 146.21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 146.21 }{ 14 } = 20.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 146.21 }{ 21 } = 13.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 146.21 }{ 24 } = 12.18 ; ;

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{ 14**2-21**2-24**2 }{ 2 * 21 * 24 } ) = 35° 27'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-14**2-24**2 }{ 2 * 14 * 24 } ) = 60° 29'28" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-14**2-21**2 }{ 2 * 21 * 14 } ) = 84° 2'43" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 146.21 }{ 29.5 } = 4.96 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 35° 27'49" } = 12.07 ; ;




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