8 13 15 triangle

Acute scalene triangle.

Sides: a = 8   b = 13   c = 15

Area: T = 51.96215242271
Perimeter: p = 36
Semiperimeter: s = 18

Angle ∠ A = α = 32.2044227504° = 32°12'15″ = 0.5622069803 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 87.7965772496° = 87°47'45″ = 1.53223252994 rad

Height: ha = 12.99903810568
Height: hb = 7.99440806503
Height: hc = 6.92882032303

Median: ma = 13.45436240471
Median: mb = 10.11218742081
Median: mc = 7.76220873481

Inradius: r = 2.88767513459
Circumradius: R = 7.50655534995

Vertex coordinates: A[15; 0] B[0; 0] C[4; 6.92882032303]
Centroid: CG[6.33333333333; 2.30994010768]
Coordinates of the circumscribed circle: U[7.5; 0.28986751346]
Coordinates of the inscribed circle: I[5; 2.88767513459]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.7965772496° = 147°47'45″ = 0.5622069803 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 92.2044227504° = 92°12'15″ = 1.53223252994 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 = 8 ; ; b = 13 ; ; c = 15 ; ;

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

p = a+b+c = 8+13+15 = 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 * (18-8)(18-13)(18-15) } ; ; T = sqrt{ 2700 } = 51.96 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 51.96 }{ 8 } = 12.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 51.96 }{ 13 } = 7.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 51.96 }{ 15 } = 6.93 ; ;

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{ 8**2-13**2-15**2 }{ 2 * 13 * 15 } ) = 32° 12'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-8**2-15**2 }{ 2 * 8 * 15 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-8**2-13**2 }{ 2 * 13 * 8 } ) = 87° 47'45" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 51.96 }{ 18 } = 2.89 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 32° 12'15" } = 7.51 ; ;




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