10 12 14 triangle

Acute scalene triangle.

Sides: a = 10   b = 12   c = 14

Area: T = 58.78877538268
Perimeter: p = 36
Semiperimeter: s = 18

Angle ∠ A = α = 44.41553085972° = 44°24'55″ = 0.77551933733 rad
Angle ∠ B = β = 57.12216504356° = 57°7'18″ = 0.99769608743 rad
Angle ∠ C = γ = 78.46330409672° = 78°27'47″ = 1.3699438406 rad

Height: ha = 11.75875507654
Height: hb = 9.79879589711
Height: hc = 8.39882505467

Median: ma = 12.04215945788
Median: mb = 10.58330052443
Median: mc = 8.54440037453

Inradius: r = 3.26659863237
Circumradius: R = 7.14443450831

Vertex coordinates: A[14; 0] B[0; 0] C[5.42985714286; 8.39882505467]
Centroid: CG[6.47661904762; 2.79994168489]
Coordinates of the circumscribed circle: U[7; 1.42988690166]
Coordinates of the inscribed circle: I[6; 3.26659863237]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.5854691403° = 135°35'5″ = 0.77551933733 rad
∠ B' = β' = 122.8788349564° = 122°52'42″ = 0.99769608743 rad
∠ C' = γ' = 101.5376959033° = 101°32'13″ = 1.3699438406 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 = 14 ; ;

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

p = a+b+c = 10+12+14 = 36 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 36 }{ 2 } = 18 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18 * (18-10)(18-12)(18-14) } ; ; T = sqrt{ 3456 } = 58.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 58.79 }{ 10 } = 11.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 58.79 }{ 12 } = 9.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 58.79 }{ 14 } = 8.4 ; ;

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**2-14**2 }{ 2 * 12 * 14 } ) = 44° 24'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-10**2-14**2 }{ 2 * 10 * 14 } ) = 57° 7'18" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14**2-10**2-12**2 }{ 2 * 12 * 10 } ) = 78° 27'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 58.79 }{ 18 } = 3.27 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 44° 24'55" } = 7.14 ; ;




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