21 23 24 triangle

Acute scalene triangle.

Sides: a = 21   b = 23   c = 24

Area: T = 220.4999433106
Perimeter: p = 68
Semiperimeter: s = 34

Angle ∠ A = α = 53.02662349852° = 53°1'34″ = 0.92554823904 rad
Angle ∠ B = β = 61.04547093922° = 61°2'41″ = 1.06554311698 rad
Angle ∠ C = γ = 65.92990556226° = 65°55'45″ = 1.15106790933 rad

Height: ha = 210.9999460101
Height: hb = 19.17438637483
Height: hc = 18.37549527588

Median: ma = 21.03297408448
Median: mb = 19.39771647413
Median: mc = 18.46661853126

Inradius: r = 6.48552774443
Circumradius: R = 13.14328909326

Vertex coordinates: A[24; 0] B[0; 0] C[10.16766666667; 18.37549527588]
Centroid: CG[11.38988888889; 6.12549842529]
Coordinates of the circumscribed circle: U[12; 5.36105579994]
Coordinates of the inscribed circle: I[11; 6.48552774443]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.9743765015° = 126°58'26″ = 0.92554823904 rad
∠ B' = β' = 118.9555290608° = 118°57'19″ = 1.06554311698 rad
∠ C' = γ' = 114.0710944377° = 114°4'15″ = 1.15106790933 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 = 21 ; ; b = 23 ; ; c = 24 ; ;

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

p = a+b+c = 21+23+24 = 68 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 68 }{ 2 } = 34 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34 * (34-21)(34-23)(34-24) } ; ; T = sqrt{ 48620 } = 220.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 220.5 }{ 21 } = 21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 220.5 }{ 23 } = 19.17 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 220.5 }{ 24 } = 18.37 ; ;

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{ 21**2-23**2-24**2 }{ 2 * 23 * 24 } ) = 53° 1'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-21**2-24**2 }{ 2 * 21 * 24 } ) = 61° 2'41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-21**2-23**2 }{ 2 * 23 * 21 } ) = 65° 55'45" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 220.5 }{ 34 } = 6.49 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 53° 1'34" } = 13.14 ; ;




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