21 23 25 triangle

Acute scalene triangle.

Sides: a = 21   b = 23   c = 25

Area: T = 225.5733020328
Perimeter: p = 69
Semiperimeter: s = 34.5

Angle ∠ A = α = 51.68438655263° = 51°41'2″ = 0.90220536236 rad
Angle ∠ B = β = 59.24109669013° = 59°14'27″ = 1.03439499245 rad
Angle ∠ C = γ = 69.07551675724° = 69°4'31″ = 1.20655891055 rad

Height: ha = 21.48331447932
Height: hb = 19.61550452459
Height: hc = 18.04658416263

Median: ma = 21.60443977005
Median: mb = 20.01987412192
Median: mc = 18.13114643645

Inradius: r = 6.53883484153
Circumradius: R = 13.38325844758

Vertex coordinates: A[25; 0] B[0; 0] C[10.74; 18.04658416263]
Centroid: CG[11.91333333333; 6.01552805421]
Coordinates of the circumscribed circle: U[12.5; 4.77994944556]
Coordinates of the inscribed circle: I[11.5; 6.53883484153]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.3166134474° = 128°18'58″ = 0.90220536236 rad
∠ B' = β' = 120.7599033099° = 120°45'33″ = 1.03439499245 rad
∠ C' = γ' = 110.9254832428° = 110°55'29″ = 1.20655891055 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 = 25 ; ;

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

p = a+b+c = 21+23+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 * (34.5-21)(34.5-23)(34.5-25) } ; ; T = sqrt{ 50883.19 } = 225.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 225.57 }{ 21 } = 21.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 225.57 }{ 23 } = 19.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 225.57 }{ 25 } = 18.05 ; ;

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**2-25**2 }{ 2 * 23 * 25 } ) = 51° 41'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-21**2-25**2 }{ 2 * 21 * 25 } ) = 59° 14'27" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-21**2-23**2 }{ 2 * 23 * 21 } ) = 69° 4'31" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 225.57 }{ 34.5 } = 6.54 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 51° 41'2" } = 13.38 ; ;




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