12 12 13 triangle

Acute isosceles triangle.

Sides: a = 12   b = 12   c = 13

Area: T = 65.5666283256
Perimeter: p = 37
Semiperimeter: s = 18.5

Angle ∠ A = α = 57.20328317042° = 57°12'10″ = 0.99883777547 rad
Angle ∠ B = β = 57.20328317042° = 57°12'10″ = 0.99883777547 rad
Angle ∠ C = γ = 65.59443365916° = 65°35'40″ = 1.14548371442 rad

Height: ha = 10.9287713876
Height: hb = 10.9287713876
Height: hc = 10.08771205009

Median: ma = 10.97772492001
Median: mb = 10.97772492001
Median: mc = 10.08771205009

Inradius: r = 3.54441234192
Circumradius: R = 7.13878149982

Vertex coordinates: A[13; 0] B[0; 0] C[6.5; 10.08771205009]
Centroid: CG[6.5; 3.36223735003]
Coordinates of the circumscribed circle: U[6.5; 2.94993055027]
Coordinates of the inscribed circle: I[6.5; 3.54441234192]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 122.7977168296° = 122°47'50″ = 0.99883777547 rad
∠ B' = β' = 122.7977168296° = 122°47'50″ = 0.99883777547 rad
∠ C' = γ' = 114.4065663408° = 114°24'20″ = 1.14548371442 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 = 12 ; ; b = 12 ; ; c = 13 ; ;

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

p = a+b+c = 12+12+13 = 37 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 37 }{ 2 } = 18.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.5 * (18.5-12)(18.5-12)(18.5-13) } ; ; T = sqrt{ 4298.94 } = 65.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 65.57 }{ 12 } = 10.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 65.57 }{ 12 } = 10.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 65.57 }{ 13 } = 10.09 ; ;

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{ 12**2-12**2-13**2 }{ 2 * 12 * 13 } ) = 57° 12'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-12**2-13**2 }{ 2 * 12 * 13 } ) = 57° 12'10" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-12**2-12**2 }{ 2 * 12 * 12 } ) = 65° 35'40" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 65.57 }{ 18.5 } = 3.54 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 57° 12'10" } = 7.14 ; ;




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