10 14 15 triangle

Acute scalene triangle.

Sides: a = 10   b = 14   c = 15

Area: T = 67.71221665582
Perimeter: p = 39
Semiperimeter: s = 19.5

Angle ∠ A = α = 40.15765122086° = 40°9'23″ = 0.70108633542 rad
Angle ∠ B = β = 64.53224398575° = 64°31'57″ = 1.12663035499 rad
Angle ∠ C = γ = 75.31110479339° = 75°18'40″ = 1.31444257496 rad

Height: ha = 13.54224333116
Height: hb = 9.67331666512
Height: hc = 9.02882888744

Median: ma = 13.62198384719
Median: mb = 10.65436378763
Median: mc = 9.57986220303

Inradius: r = 3.47224187979
Circumradius: R = 7.75334072041

Vertex coordinates: A[15; 0] B[0; 0] C[4.3; 9.02882888744]
Centroid: CG[6.43333333333; 3.00994296248]
Coordinates of the circumscribed circle: U[7.5; 1.9666042541]
Coordinates of the inscribed circle: I[5.5; 3.47224187979]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.8433487791° = 139°50'37″ = 0.70108633542 rad
∠ B' = β' = 115.4687560142° = 115°28'3″ = 1.12663035499 rad
∠ C' = γ' = 104.6898952066° = 104°41'20″ = 1.31444257496 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 = 14 ; ; c = 15 ; ;

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

p = a+b+c = 10+14+15 = 39 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 39 }{ 2 } = 19.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.5 * (19.5-10)(19.5-14)(19.5-15) } ; ; T = sqrt{ 4584.94 } = 67.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 67.71 }{ 10 } = 13.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 67.71 }{ 14 } = 9.67 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 67.71 }{ 15 } = 9.03 ; ;

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**2-14**2-15**2 }{ 2 * 14 * 15 } ) = 40° 9'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 14**2-10**2-15**2 }{ 2 * 10 * 15 } ) = 64° 31'57" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-10**2-14**2 }{ 2 * 14 * 10 } ) = 75° 18'40" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 67.71 }{ 19.5 } = 3.47 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 40° 9'23" } = 7.75 ; ;




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