15 18 22 triangle

Acute scalene triangle.

Sides: a = 15   b = 18   c = 22

Area: T = 134.0188422241
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 42.59988128925° = 42°35'56″ = 0.74334895424 rad
Angle ∠ B = β = 54.31546652873° = 54°18'53″ = 0.94879697414 rad
Angle ∠ C = γ = 83.08765218202° = 83°5'11″ = 1.45501333698 rad

Height: ha = 17.86991229655
Height: hb = 14.89109358046
Height: hc = 12.1833492931

Median: ma = 18.6488056199
Median: mb = 16.53878354085
Median: mc = 12.39895116934

Inradius: r = 4.87333971724
Circumradius: R = 11.08105662025

Vertex coordinates: A[22; 0] B[0; 0] C[8.75; 12.1833492931]
Centroid: CG[10.25; 4.06111643103]
Coordinates of the circumscribed circle: U[11; 1.33437718577]
Coordinates of the inscribed circle: I[9.5; 4.87333971724]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.4011187108° = 137°24'4″ = 0.74334895424 rad
∠ B' = β' = 125.6855334713° = 125°41'7″ = 0.94879697414 rad
∠ C' = γ' = 96.91334781798° = 96°54'49″ = 1.45501333698 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 = 15 ; ; b = 18 ; ; c = 22 ; ;

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

p = a+b+c = 15+18+22 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-15)(27.5-18)(27.5-22) } ; ; T = sqrt{ 17960.94 } = 134.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 134.02 }{ 15 } = 17.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 134.02 }{ 18 } = 14.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 134.02 }{ 22 } = 12.18 ; ;

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{ 15**2-18**2-22**2 }{ 2 * 18 * 22 } ) = 42° 35'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-15**2-22**2 }{ 2 * 15 * 22 } ) = 54° 18'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-15**2-18**2 }{ 2 * 18 * 15 } ) = 83° 5'11" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 134.02 }{ 27.5 } = 4.87 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 42° 35'56" } = 11.08 ; ;




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