10 12 18 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 12   c = 18

Area: T = 56.56985424949
Perimeter: p = 40
Semiperimeter: s = 20

Angle ∠ A = α = 31.58663380965° = 31°35'11″ = 0.55112855984 rad
Angle ∠ B = β = 38.9422441269° = 38°56'33″ = 0.68796738189 rad
Angle ∠ C = γ = 109.4711220634° = 109°28'16″ = 1.91106332362 rad

Height: ha = 11.3143708499
Height: hb = 9.42880904158
Height: hc = 6.28553936105

Median: ma = 14.45768322948
Median: mb = 13.26664991614
Median: mc = 6.40331242374

Inradius: r = 2.82884271247
Circumradius: R = 9.5465941546

Vertex coordinates: A[18; 0] B[0; 0] C[7.77877777778; 6.28553936105]
Centroid: CG[8.59325925926; 2.09551312035]
Coordinates of the circumscribed circle: U[9; -3.18219805153]
Coordinates of the inscribed circle: I[8; 2.82884271247]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.4143661903° = 148°24'49″ = 0.55112855984 rad
∠ B' = β' = 141.0587558731° = 141°3'27″ = 0.68796738189 rad
∠ C' = γ' = 70.52987793655° = 70°31'44″ = 1.91106332362 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 = 10 ; ; b = 12 ; ; c = 18 ; ;

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

p = a+b+c = 10+12+18 = 40 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40 }{ 2 } = 20 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20 * (20-10)(20-12)(20-18) } ; ; T = sqrt{ 3200 } = 56.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 56.57 }{ 10 } = 11.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 56.57 }{ 12 } = 9.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 56.57 }{ 18 } = 6.29 ; ;

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{ 10**2-12**2-18**2 }{ 2 * 12 * 18 } ) = 31° 35'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-10**2-18**2 }{ 2 * 10 * 18 } ) = 38° 56'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 18**2-10**2-12**2 }{ 2 * 12 * 10 } ) = 109° 28'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 56.57 }{ 20 } = 2.83 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 31° 35'11" } = 9.55 ; ;




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