10 11 13 triangle

Acute scalene triangle.

Sides: a = 10   b = 11   c = 13

Area: T = 53.44215568635
Perimeter: p = 34
Semiperimeter: s = 17

Angle ∠ A = α = 48.36986204606° = 48°22'7″ = 0.84441916817 rad
Angle ∠ B = β = 55.30333976437° = 55°18'12″ = 0.96552263764 rad
Angle ∠ C = γ = 76.32879818956° = 76°19'41″ = 1.33221745955 rad

Height: ha = 10.68883113727
Height: hb = 9.71766467025
Height: hc = 8.2221777979

Median: ma = 10.95444511501
Median: mb = 10.21102889283
Median: mc = 8.26113558209

Inradius: r = 3.1443620992
Circumradius: R = 6.69895506228

Vertex coordinates: A[13; 0] B[0; 0] C[5.69223076923; 8.2221777979]
Centroid: CG[6.23107692308; 2.74105926597]
Coordinates of the circumscribed circle: U[6.5; 1.58111665108]
Coordinates of the inscribed circle: I[6; 3.1443620992]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.6311379539° = 131°37'53″ = 0.84441916817 rad
∠ B' = β' = 124.6976602356° = 124°41'48″ = 0.96552263764 rad
∠ C' = γ' = 103.6722018104° = 103°40'19″ = 1.33221745955 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 = 11 ; ; c = 13 ; ;

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

p = a+b+c = 10+11+13 = 34 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 34 }{ 2 } = 17 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17 * (17-10)(17-11)(17-13) } ; ; T = sqrt{ 2856 } = 53.44 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 53.44 }{ 10 } = 10.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 53.44 }{ 11 } = 9.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 53.44 }{ 13 } = 8.22 ; ;

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**2-11**2-13**2 }{ 2 * 11 * 13 } ) = 48° 22'7" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11**2-10**2-13**2 }{ 2 * 10 * 13 } ) = 55° 18'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-10**2-11**2 }{ 2 * 11 * 10 } ) = 76° 19'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 53.44 }{ 17 } = 3.14 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 48° 22'7" } = 6.69 ; ;




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