18 21 25 triangle

Acute scalene triangle.

Sides: a = 18   b = 21   c = 25

Area: T = 185.731098826
Perimeter: p = 64
Semiperimeter: s = 32

Angle ∠ A = α = 45.03656507165° = 45°2'8″ = 0.78660203858 rad
Angle ∠ B = β = 55.63662785693° = 55°38'11″ = 0.97110362446 rad
Angle ∠ C = γ = 79.32880707142° = 79°19'41″ = 1.38545360232 rad

Height: ha = 20.63767764733
Height: hb = 17.68986655486
Height: hc = 14.85884790608

Median: ma = 21.26602916255
Median: mb = 19.0855334684
Median: mc = 15.04216089565

Inradius: r = 5.80440933831
Circumradius: R = 12.72200098494

Vertex coordinates: A[25; 0] B[0; 0] C[10.16; 14.85884790608]
Centroid: CG[11.72; 4.95328263536]
Coordinates of the circumscribed circle: U[12.5; 2.35655573795]
Coordinates of the inscribed circle: I[11; 5.80440933831]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.9644349284° = 134°57'52″ = 0.78660203858 rad
∠ B' = β' = 124.3643721431° = 124°21'49″ = 0.97110362446 rad
∠ C' = γ' = 100.6721929286° = 100°40'19″ = 1.38545360232 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 = 21 ; ; c = 25 ; ;

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

p = a+b+c = 18+21+25 = 64 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 64 }{ 2 } = 32 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32 * (32-18)(32-21)(32-25) } ; ; T = sqrt{ 34496 } = 185.73 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 185.73 }{ 18 } = 20.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 185.73 }{ 21 } = 17.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 185.73 }{ 25 } = 14.86 ; ;

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**2-21**2-25**2 }{ 2 * 21 * 25 } ) = 45° 2'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-18**2-25**2 }{ 2 * 18 * 25 } ) = 55° 38'11" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-18**2-21**2 }{ 2 * 21 * 18 } ) = 79° 19'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 185.73 }{ 32 } = 5.8 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18 }{ 2 * sin 45° 2'8" } = 12.72 ; ;




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