20 24 25 triangle

Acute scalene triangle.

Sides: a = 20   b = 24   c = 25

Area: T = 223.3832939143
Perimeter: p = 69
Semiperimeter: s = 34.5

Angle ∠ A = α = 48.12655944006° = 48°7'32″ = 0.84399500768 rad
Angle ∠ B = β = 63.32204569276° = 63°19'14″ = 1.10551504573 rad
Angle ∠ C = γ = 68.55439486718° = 68°33'14″ = 1.19664921196 rad

Height: ha = 22.33882939143
Height: hb = 18.61552449286
Height: hc = 17.87106351314

Median: ma = 22.37218573212
Median: mb = 19.19663538205
Median: mc = 18.21440056001

Inradius: r = 6.47548678012
Circumradius: R = 13.43298528416

Vertex coordinates: A[25; 0] B[0; 0] C[8.98; 17.87106351314]
Centroid: CG[11.32766666667; 5.95768783771]
Coordinates of the circumscribed circle: U[12.5; 4.91102899452]
Coordinates of the inscribed circle: I[10.5; 6.47548678012]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.8744405599° = 131°52'28″ = 0.84399500768 rad
∠ B' = β' = 116.6879543072° = 116°40'46″ = 1.10551504573 rad
∠ C' = γ' = 111.4466051328° = 111°26'46″ = 1.19664921196 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 = 20 ; ; b = 24 ; ; c = 25 ; ;

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

p = a+b+c = 20+24+25 = 69 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 69 }{ 2 } = 34.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.5 * (34.5-20)(34.5-24)(34.5-25) } ; ; T = sqrt{ 49899.94 } = 223.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 223.38 }{ 20 } = 22.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 223.38 }{ 24 } = 18.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 223.38 }{ 25 } = 17.87 ; ;

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{ 20**2-24**2-25**2 }{ 2 * 24 * 25 } ) = 48° 7'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-20**2-25**2 }{ 2 * 20 * 25 } ) = 63° 19'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-20**2-24**2 }{ 2 * 24 * 20 } ) = 68° 33'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 223.38 }{ 34.5 } = 6.47 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 48° 7'32" } = 13.43 ; ;




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