10 12 13 triangle

Acute scalene triangle.

Sides: a = 10   b = 12   c = 13

Area: T = 56.99550655759
Perimeter: p = 35
Semiperimeter: s = 17.5

Angle ∠ A = α = 46.94656106092° = 46°56'44″ = 0.81993554745 rad
Angle ∠ B = β = 61.26443462551° = 61°15'52″ = 1.06992645562 rad
Angle ∠ C = γ = 71.79900431357° = 71°47'24″ = 1.25329726229 rad

Height: ha = 11.39990131152
Height: hb = 9.4999177596
Height: hc = 8.76884716271

Median: ma = 11.46773449412
Median: mb = 9.92547166206
Median: mc = 8.93302855497

Inradius: r = 3.25768608901
Circumradius: R = 6.84326976276

Vertex coordinates: A[13; 0] B[0; 0] C[4.80876923077; 8.76884716271]
Centroid: CG[5.93658974359; 2.92328238757]
Coordinates of the circumscribed circle: U[6.5; 2.13883430086]
Coordinates of the inscribed circle: I[5.5; 3.25768608901]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.0544389391° = 133°3'16″ = 0.81993554745 rad
∠ B' = β' = 118.7365653745° = 118°44'8″ = 1.06992645562 rad
∠ C' = γ' = 108.2109956864° = 108°12'36″ = 1.25329726229 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 = 13 ; ;

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

p = a+b+c = 10+12+13 = 35 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 35 }{ 2 } = 17.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.5 * (17.5-10)(17.5-12)(17.5-13) } ; ; T = sqrt{ 3248.44 } = 57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 57 }{ 10 } = 11.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 57 }{ 12 } = 9.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 57 }{ 13 } = 8.77 ; ;

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**2-13**2 }{ 2 * 12 * 13 } ) = 46° 56'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-10**2-13**2 }{ 2 * 10 * 13 } ) = 61° 15'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-10**2-12**2 }{ 2 * 12 * 10 } ) = 71° 47'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 57 }{ 17.5 } = 3.26 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 46° 56'44" } = 6.84 ; ;




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