18 20 24 triangle

Acute scalene triangle.

Sides: a = 18   b = 20   c = 24

Area: T = 176.1566180703
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 47.22114422911° = 47°13'17″ = 0.82441696455 rad
Angle ∠ B = β = 54.64105803778° = 54°38'26″ = 0.95436580328 rad
Angle ∠ C = γ = 78.13879773311° = 78°8'17″ = 1.36437649753 rad

Height: ha = 19.5732908967
Height: hb = 17.61656180703
Height: hc = 14.68796817253

Median: ma = 20.17442410018
Median: mb = 18.70882869339
Median: mc = 14.76548230602

Inradius: r = 5.6822457442
Circumradius: R = 12.26218462286

Vertex coordinates: A[24; 0] B[0; 0] C[10.41766666667; 14.68796817253]
Centroid: CG[11.47222222222; 4.89332272418]
Coordinates of the circumscribed circle: U[12; 2.52204906137]
Coordinates of the inscribed circle: I[11; 5.6822457442]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.7798557709° = 132°46'43″ = 0.82441696455 rad
∠ B' = β' = 125.3599419622° = 125°21'34″ = 0.95436580328 rad
∠ C' = γ' = 101.8622022669° = 101°51'43″ = 1.36437649753 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 = 18 ; ; b = 20 ; ; c = 24 ; ;

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

p = a+b+c = 18+20+24 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-18)(31-20)(31-24) } ; ; T = sqrt{ 31031 } = 176.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 176.16 }{ 18 } = 19.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 176.16 }{ 20 } = 17.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 176.16 }{ 24 } = 14.68 ; ;

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{ 18**2-20**2-24**2 }{ 2 * 20 * 24 } ) = 47° 13'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-18**2-24**2 }{ 2 * 18 * 24 } ) = 54° 38'26" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-18**2-20**2 }{ 2 * 20 * 18 } ) = 78° 8'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 176.16 }{ 31 } = 5.68 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 47° 13'17" } = 12.26 ; ;




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