12 12 14 triangle

Acute isosceles triangle.

Sides: a = 12   b = 12   c = 14

Area: T = 68.22875604137
Perimeter: p = 38
Semiperimeter: s = 19

Angle ∠ A = α = 54.31546652873° = 54°18'53″ = 0.94879697414 rad
Angle ∠ B = β = 54.31546652873° = 54°18'53″ = 0.94879697414 rad
Angle ∠ C = γ = 71.37106694253° = 71°22'14″ = 1.24656531708 rad

Height: ha = 11.37112600689
Height: hb = 11.37112600689
Height: hc = 9.74767943448

Median: ma = 11.57658369028
Median: mb = 11.57658369028
Median: mc = 9.74767943448

Inradius: r = 3.59109242323
Circumradius: R = 7.3877044135

Vertex coordinates: A[14; 0] B[0; 0] C[7; 9.74767943448]
Centroid: CG[7; 3.24989314483]
Coordinates of the circumscribed circle: U[7; 2.36597502098]
Coordinates of the inscribed circle: I[7; 3.59109242323]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.6855334713° = 125°41'7″ = 0.94879697414 rad
∠ B' = β' = 125.6855334713° = 125°41'7″ = 0.94879697414 rad
∠ C' = γ' = 108.6299330575° = 108°37'46″ = 1.24656531708 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 = 14 ; ;

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

p = a+b+c = 12+12+14 = 38 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38 }{ 2 } = 19 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19 * (19-12)(19-12)(19-14) } ; ; T = sqrt{ 4655 } = 68.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 68.23 }{ 12 } = 11.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 68.23 }{ 12 } = 11.37 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 68.23 }{ 14 } = 9.75 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 12**2+14**2-12**2 }{ 2 * 12 * 14 } ) = 54° 18'53" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12**2+14**2-12**2 }{ 2 * 12 * 14 } ) = 54° 18'53" ; ; gamma = 180° - alpha - beta = 180° - 54° 18'53" - 54° 18'53" = 71° 22'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 68.23 }{ 19 } = 3.59 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12 }{ 2 * sin 54° 18'53" } = 7.39 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 14**2 - 12**2 } }{ 2 } = 11.576 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 14**2+2 * 12**2 - 12**2 } }{ 2 } = 11.576 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 12**2 - 14**2 } }{ 2 } = 9.747 ; ;
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