12 12 15 triangle

Acute isosceles triangle.

Sides: a = 12   b = 12   c = 15

Area: T = 70.2566227482
Perimeter: p = 39
Semiperimeter: s = 19.5

Angle ∠ A = α = 51.31878125465° = 51°19'4″ = 0.89656647939 rad
Angle ∠ B = β = 51.31878125465° = 51°19'4″ = 0.89656647939 rad
Angle ∠ C = γ = 77.3644374907° = 77°21'52″ = 1.35502630659 rad

Height: ha = 11.7099371247
Height: hb = 11.7099371247
Height: hc = 9.36774969976

Median: ma = 12.1866057607
Median: mb = 12.1866057607
Median: mc = 9.36774969976

Inradius: r = 3.60328834606
Circumradius: R = 7.68661513826

Vertex coordinates: A[15; 0] B[0; 0] C[7.5; 9.36774969976]
Centroid: CG[7.5; 3.12224989992]
Coordinates of the circumscribed circle: U[7.5; 1.6811345615]
Coordinates of the inscribed circle: I[7.5; 3.60328834606]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.6822187453° = 128°40'56″ = 0.89656647939 rad
∠ B' = β' = 128.6822187453° = 128°40'56″ = 0.89656647939 rad
∠ C' = γ' = 102.6365625093° = 102°38'8″ = 1.35502630659 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 = 15 ; ;

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

p = a+b+c = 12+12+15 = 39 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 39 }{ 2 } = 19.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.5 * (19.5-12)(19.5-12)(19.5-15) } ; ; T = sqrt{ 4935.94 } = 70.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 70.26 }{ 12 } = 11.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 70.26 }{ 12 } = 11.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 70.26 }{ 15 } = 9.37 ; ;

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**2-15**2 }{ 2 * 12 * 15 } ) = 51° 19'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-12**2-15**2 }{ 2 * 12 * 15 } ) = 51° 19'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-12**2-12**2 }{ 2 * 12 * 12 } ) = 77° 21'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 70.26 }{ 19.5 } = 3.6 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 51° 19'4" } = 7.69 ; ;




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