12 15 15 triangle

Acute isosceles triangle.

Sides: a = 12   b = 15   c = 15

Area: T = 82.48663625092
Perimeter: p = 42
Semiperimeter: s = 21

Angle ∠ A = α = 47.15663569564° = 47°9'23″ = 0.82330336921 rad
Angle ∠ B = β = 66.42218215218° = 66°25'19″ = 1.15992794807 rad
Angle ∠ C = γ = 66.42218215218° = 66°25'19″ = 1.15992794807 rad

Height: ha = 13.74877270849
Height: hb = 10.99881816679
Height: hc = 10.99881816679

Median: ma = 13.74877270849
Median: mb = 11.32547516529
Median: mc = 11.32547516529

Inradius: r = 3.92879220242
Circumradius: R = 8.18331708838

Vertex coordinates: A[15; 0] B[0; 0] C[4.8; 10.99881816679]
Centroid: CG[6.6; 3.6666060556]
Coordinates of the circumscribed circle: U[7.5; 3.27332683535]
Coordinates of the inscribed circle: I[6; 3.92879220242]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.8443643044° = 132°50'37″ = 0.82330336921 rad
∠ B' = β' = 113.5788178478° = 113°34'41″ = 1.15992794807 rad
∠ C' = γ' = 113.5788178478° = 113°34'41″ = 1.15992794807 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 = 15 ; ; c = 15 ; ;

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

p = a+b+c = 12+15+15 = 42 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 42 }{ 2 } = 21 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21 * (21-12)(21-15)(21-15) } ; ; T = sqrt{ 6804 } = 82.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 82.49 }{ 12 } = 13.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 82.49 }{ 15 } = 11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 82.49 }{ 15 } = 11 ; ;

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**2-15**2-15**2 }{ 2 * 15 * 15 } ) = 47° 9'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-12**2-15**2 }{ 2 * 12 * 15 } ) = 66° 25'19" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-12**2-15**2 }{ 2 * 15 * 12 } ) = 66° 25'19" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 82.49 }{ 21 } = 3.93 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 47° 9'23" } = 8.18 ; ;




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