12 30 30 triangle

Acute isosceles triangle.

Sides: a = 12   b = 30   c = 30

Area: T = 176.363326148
Perimeter: p = 72
Semiperimeter: s = 36

Angle ∠ A = α = 23.07439180656° = 23°4'26″ = 0.40327158416 rad
Angle ∠ B = β = 78.46330409672° = 78°27'47″ = 1.3699438406 rad
Angle ∠ C = γ = 78.46330409672° = 78°27'47″ = 1.3699438406 rad

Height: ha = 29.39438769134
Height: hb = 11.75875507654
Height: hc = 11.75875507654

Median: ma = 29.39438769134
Median: mb = 17.23436879396
Median: mc = 17.23436879396

Inradius: r = 4.89989794856
Circumradius: R = 15.30993108924

Vertex coordinates: A[30; 0] B[0; 0] C[2.4; 11.75875507654]
Centroid: CG[10.8; 3.91991835885]
Coordinates of the circumscribed circle: U[15; 3.06218621785]
Coordinates of the inscribed circle: I[6; 4.89989794856]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.9266081934° = 156°55'34″ = 0.40327158416 rad
∠ B' = β' = 101.5376959033° = 101°32'13″ = 1.3699438406 rad
∠ C' = γ' = 101.5376959033° = 101°32'13″ = 1.3699438406 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 = 30 ; ; c = 30 ; ;

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

p = a+b+c = 12+30+30 = 72 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72 }{ 2 } = 36 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36 * (36-12)(36-30)(36-30) } ; ; T = sqrt{ 31104 } = 176.36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 176.36 }{ 12 } = 29.39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 176.36 }{ 30 } = 11.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 176.36 }{ 30 } = 11.76 ; ;

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**2-30**2-30**2 }{ 2 * 30 * 30 } ) = 23° 4'26" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 30**2-12**2-30**2 }{ 2 * 12 * 30 } ) = 78° 27'47" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-12**2-30**2 }{ 2 * 30 * 12 } ) = 78° 27'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 176.36 }{ 36 } = 4.9 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 23° 4'26" } = 15.31 ; ;




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