15 16 22 triangle

Obtuse scalene triangle.

Sides: a = 15   b = 16   c = 22

Area: T = 119.9987656227
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 42.98548430555° = 42°59'5″ = 0.75502270398 rad
Angle ∠ B = β = 46.65770559912° = 46°39'25″ = 0.81443192463 rad
Angle ∠ C = γ = 90.35881009534° = 90°21'29″ = 1.57770463675 rad

Height: ha = 165.9996874969
Height: hb = 154.9997070284
Height: hc = 10.90988778388

Median: ma = 17.71329895839
Median: mb = 17.04440605491
Median: mc = 10.93216055545

Inradius: r = 4.52882134425
Circumradius: R = 111.00021485

Vertex coordinates: A[22; 0] B[0; 0] C[10.29554545455; 10.90988778388]
Centroid: CG[10.76551515152; 3.63662926129]
Coordinates of the circumscribed circle: U[11; -0.06987513428]
Coordinates of the inscribed circle: I[10.5; 4.52882134425]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.0155156945° = 137°55″ = 0.75502270398 rad
∠ B' = β' = 133.3432944009° = 133°20'35″ = 0.81443192463 rad
∠ C' = γ' = 89.64218990466° = 89°38'31″ = 1.57770463675 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 = 16 ; ; c = 22 ; ;

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

p = a+b+c = 15+16+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 * (26.5-15)(26.5-16)(26.5-22) } ; ; T = sqrt{ 14399.44 } = 120 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 120 }{ 15 } = 16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 120 }{ 16 } = 15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 120 }{ 22 } = 10.91 ; ;

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**2-16**2-22**2 }{ 2 * 16 * 22 } ) = 42° 59'5" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-15**2-22**2 }{ 2 * 15 * 22 } ) = 46° 39'25" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-15**2-16**2 }{ 2 * 16 * 15 } ) = 90° 21'29" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 120 }{ 26.5 } = 4.53 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 42° 59'5" } = 11 ; ;




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