13 18 22 triangle

Acute scalene triangle.

Sides: a = 13   b = 18   c = 22

Area: T = 116.9788363384
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 36.21437860673° = 36°12'50″ = 0.63220498015 rad
Angle ∠ B = β = 54.8888123772° = 54°53'17″ = 0.95879784801 rad
Angle ∠ C = γ = 88.89880901607° = 88°53'53″ = 1.5521564372 rad

Height: ha = 17.99766712898
Height: hb = 12.99875959316
Height: hc = 10.63443966713

Median: ma = 19.02197266016
Median: mb = 15.6688439616
Median: mc = 11.20326782512

Inradius: r = 4.41442778635
Circumradius: R = 11.0022034588

Vertex coordinates: A[22; 0] B[0; 0] C[7.47772727273; 10.63443966713]
Centroid: CG[9.82657575758; 3.54547988904]
Coordinates of the circumscribed circle: U[11; 0.21215775882]
Coordinates of the inscribed circle: I[8.5; 4.41442778635]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.7866213933° = 143°47'10″ = 0.63220498015 rad
∠ B' = β' = 125.1121876228° = 125°6'43″ = 0.95879784801 rad
∠ C' = γ' = 91.10219098393° = 91°6'7″ = 1.5521564372 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 = 13 ; ; b = 18 ; ; c = 22 ; ;

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

p = a+b+c = 13+18+22 = 53 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53 }{ 2 } = 26.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.5 * (26.5-13)(26.5-18)(26.5-22) } ; ; T = sqrt{ 13683.94 } = 116.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 116.98 }{ 13 } = 18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 116.98 }{ 18 } = 13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 116.98 }{ 22 } = 10.63 ; ;

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{ 13**2-18**2-22**2 }{ 2 * 18 * 22 } ) = 36° 12'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-13**2-22**2 }{ 2 * 13 * 22 } ) = 54° 53'17" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-13**2-18**2 }{ 2 * 18 * 13 } ) = 88° 53'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 116.98 }{ 26.5 } = 4.41 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 36° 12'50" } = 11 ; ;




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