15 16 18 triangle

Acute scalene triangle.

Sides: a = 15   b = 16   c = 18

Area: T = 113.3999459875
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 51.95221438479° = 51°57'8″ = 0.90767359636 rad
Angle ∠ B = β = 57.14396971359° = 57°8'23″ = 0.99772758486 rad
Angle ∠ C = γ = 70.90881590162° = 70°54'29″ = 1.23875808414 rad

Height: ha = 15.12199279834
Height: hb = 14.17549324844
Height: hc = 12.65999399861

Median: ma = 15.28988848514
Median: mb = 14.50986181285
Median: mc = 12.62993309403

Inradius: r = 4.62985493827
Circumradius: R = 9.5243854886

Vertex coordinates: A[18; 0] B[0; 0] C[8.13988888889; 12.65999399861]
Centroid: CG[8.7132962963; 4.21999799954]
Coordinates of the circumscribed circle: U[9; 3.11550942023]
Coordinates of the inscribed circle: I[8.5; 4.62985493827]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.0487856152° = 128°2'52″ = 0.90767359636 rad
∠ B' = β' = 122.8660302864° = 122°51'37″ = 0.99772758486 rad
∠ C' = γ' = 109.0921840984° = 109°5'31″ = 1.23875808414 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 = 18 ; ;

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

p = a+b+c = 15+16+18 = 49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49 }{ 2 } = 24.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.5 * (24.5-15)(24.5-16)(24.5-18) } ; ; T = sqrt{ 12859.44 } = 113.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 113.4 }{ 15 } = 15.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 113.4 }{ 16 } = 14.17 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 113.4 }{ 18 } = 12.6 ; ;

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**2-18**2 }{ 2 * 16 * 18 } ) = 51° 57'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-15**2-18**2 }{ 2 * 15 * 18 } ) = 57° 8'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 18**2-15**2-16**2 }{ 2 * 16 * 15 } ) = 70° 54'29" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 113.4 }{ 24.5 } = 4.63 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 51° 57'8" } = 9.52 ; ;




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