13 14 15 triangle

Acute scalene triangle.

Sides: a = 13   b = 14   c = 15

Area: T = 84
Perimeter: p = 42
Semiperimeter: s = 21

Angle ∠ A = α = 53.13301023542° = 53°7'48″ = 0.9277295218 rad
Angle ∠ B = β = 59.49897625939° = 59°29'23″ = 1.03882922285 rad
Angle ∠ C = γ = 67.3880135052° = 67°22'49″ = 1.17660052071 rad

Height: ha = 12.92330769231
Height: hb = 12
Height: hc = 11.2

Median: ma = 12.97111217711
Median: mb = 12.16655250606
Median: mc = 11.23661025271

Inradius: r = 4
Circumradius: R = 8.125

Vertex coordinates: A[15; 0] B[0; 0] C[6.6; 11.2]
Centroid: CG[7.2; 3.73333333333]
Coordinates of the circumscribed circle: U[7.5; 3.125]
Coordinates of the inscribed circle: I[7; 4]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.8769897646° = 126°52'12″ = 0.9277295218 rad
∠ B' = β' = 120.5110237406° = 120°30'37″ = 1.03882922285 rad
∠ C' = γ' = 112.6219864948° = 112°37'11″ = 1.17660052071 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 = 13 ; ; b = 14 ; ; c = 15 ; ;

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

p = a+b+c = 13+14+15 = 42 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 42 }{ 2 } = 21 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21 * (21-13)(21-14)(21-15) } ; ; T = sqrt{ 7056 } = 84 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 84 }{ 13 } = 12.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 84 }{ 14 } = 12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 84 }{ 15 } = 11.2 ; ;

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{ 13**2-14**2-15**2 }{ 2 * 14 * 15 } ) = 53° 7'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 14**2-13**2-15**2 }{ 2 * 13 * 15 } ) = 59° 29'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-13**2-14**2 }{ 2 * 14 * 13 } ) = 67° 22'49" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 84 }{ 21 } = 4 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 53° 7'48" } = 8.13 ; ;




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