21 22 25 triangle

Acute scalene triangle.

Sides: a = 21   b = 22   c = 25

Area: T = 218.4865697472
Perimeter: p = 68
Semiperimeter: s = 34

Angle ∠ A = α = 52.60774364381° = 52°36'27″ = 0.91881729769 rad
Angle ∠ B = β = 56.33884693605° = 56°20'18″ = 0.98332917859 rad
Angle ∠ C = γ = 71.05440942014° = 71°3'15″ = 1.24401278908 rad

Height: ha = 20.8088161664
Height: hb = 19.86223361339
Height: hc = 17.47988557978

Median: ma = 21.07772389084
Median: mb = 20.29877831302
Median: mc = 17.5

Inradius: r = 6.42660499257
Circumradius: R = 13.21659680629

Vertex coordinates: A[25; 0] B[0; 0] C[11.64; 17.47988557978]
Centroid: CG[12.21333333333; 5.82662852659]
Coordinates of the circumscribed circle: U[12.5; 4.29108987217]
Coordinates of the inscribed circle: I[12; 6.42660499257]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.3932563562° = 127°23'33″ = 0.91881729769 rad
∠ B' = β' = 123.662153064° = 123°39'42″ = 0.98332917859 rad
∠ C' = γ' = 108.9465905799° = 108°56'45″ = 1.24401278908 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 = 22 ; ; c = 25 ; ;

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

p = a+b+c = 21+22+25 = 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)(34-22)(34-25) } ; ; T = sqrt{ 47736 } = 218.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 218.49 }{ 21 } = 20.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 218.49 }{ 22 } = 19.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 218.49 }{ 25 } = 17.48 ; ;

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**2-22**2-25**2 }{ 2 * 22 * 25 } ) = 52° 36'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-21**2-25**2 }{ 2 * 21 * 25 } ) = 56° 20'18" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-21**2-22**2 }{ 2 * 22 * 21 } ) = 71° 3'15" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 218.49 }{ 34 } = 6.43 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 52° 36'27" } = 13.22 ; ;




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