15 18 20 triangle

Acute scalene triangle.

Sides: a = 15   b = 18   c = 20

Area: T = 129.7599151893
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 46.1287528069° = 46°7'39″ = 0.80550772406 rad
Angle ∠ B = β = 59.89896728258° = 59°53'23″ = 1.04552719788 rad
Angle ∠ C = γ = 73.98327991052° = 73°58'58″ = 1.29112434342 rad

Height: ha = 17.30112202524
Height: hb = 14.41876835437
Height: hc = 12.97659151893

Median: ma = 17.486570845
Median: mb = 15.21551240547
Median: mc = 13.21098448136

Inradius: r = 4.89765717695
Circumradius: R = 10.40438904409

Vertex coordinates: A[20; 0] B[0; 0] C[7.525; 12.97659151893]
Centroid: CG[9.175; 4.32553050631]
Coordinates of the circumscribed circle: U[10; 2.87107031031]
Coordinates of the inscribed circle: I[8.5; 4.89765717695]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.8722471931° = 133°52'21″ = 0.80550772406 rad
∠ B' = β' = 120.1110327174° = 120°6'37″ = 1.04552719788 rad
∠ C' = γ' = 106.0177200895° = 106°1'2″ = 1.29112434342 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 = 20 ; ;

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

p = a+b+c = 15+18+20 = 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)(26.5-18)(26.5-20) } ; ; T = sqrt{ 16837.44 } = 129.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 129.76 }{ 15 } = 17.3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 129.76 }{ 18 } = 14.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 129.76 }{ 20 } = 12.98 ; ;

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**2-20**2 }{ 2 * 18 * 20 } ) = 46° 7'39" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-15**2-20**2 }{ 2 * 15 * 20 } ) = 59° 53'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-15**2-18**2 }{ 2 * 18 * 15 } ) = 73° 58'58" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 129.76 }{ 26.5 } = 4.9 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 46° 7'39" } = 10.4 ; ;




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