21 24 25 triangle

Acute scalene triangle.

Sides: a = 21   b = 24   c = 25

Area: T = 232.1643735325
Perimeter: p = 70
Semiperimeter: s = 35

Angle ∠ A = α = 50.70435197608° = 50°42'13″ = 0.88549433622 rad
Angle ∠ B = β = 62.18218607153° = 62°10'55″ = 1.08552782045 rad
Angle ∠ C = γ = 67.11546195238° = 67°6'53″ = 1.17113710869 rad

Height: ha = 22.11108319357
Height: hb = 19.34769779437
Height: hc = 18.5733098826

Median: ma = 22.14215898255
Median: mb = 19.72330829233
Median: mc = 18.76883243791

Inradius: r = 6.63332495807
Circumradius: R = 13.5688010506

Vertex coordinates: A[25; 0] B[0; 0] C[9.8; 18.5733098826]
Centroid: CG[11.6; 6.1911032942]
Coordinates of the circumscribed circle: U[12.5; 5.27664485301]
Coordinates of the inscribed circle: I[11; 6.63332495807]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.2966480239° = 129°17'47″ = 0.88549433622 rad
∠ B' = β' = 117.8188139285° = 117°49'5″ = 1.08552782045 rad
∠ C' = γ' = 112.8855380476° = 112°53'7″ = 1.17113710869 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 = 24 ; ; c = 25 ; ;

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

p = a+b+c = 21+24+25 = 70 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 70 }{ 2 } = 35 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35 * (35-21)(35-24)(35-25) } ; ; T = sqrt{ 53900 } = 232.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 232.16 }{ 21 } = 22.11 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 232.16 }{ 24 } = 19.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 232.16 }{ 25 } = 18.57 ; ;

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**2-24**2-25**2 }{ 2 * 24 * 25 } ) = 50° 42'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-21**2-25**2 }{ 2 * 21 * 25 } ) = 62° 10'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-21**2-24**2 }{ 2 * 24 * 21 } ) = 67° 6'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 232.16 }{ 35 } = 6.63 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 50° 42'13" } = 13.57 ; ;




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