15 15 23 triangle

Obtuse isosceles triangle.

Sides: a = 15   b = 15   c = 23

Area: T = 110.7532821634
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 39.94545051898° = 39°56'40″ = 0.69771631336 rad
Angle ∠ B = β = 39.94545051898° = 39°56'40″ = 0.69771631336 rad
Angle ∠ C = γ = 100.111098962° = 100°6'40″ = 1.74772663863 rad

Height: ha = 14.76770428846
Height: hb = 14.76770428846
Height: hc = 9.63106801421

Median: ma = 17.90994946886
Median: mb = 17.90994946886
Median: mc = 9.63106801421

Inradius: r = 4.17993517598
Circumradius: R = 11.68114179622

Vertex coordinates: A[23; 0] B[0; 0] C[11.5; 9.63106801421]
Centroid: CG[11.5; 3.2110226714]
Coordinates of the circumscribed circle: U[11.5; -2.051073782]
Coordinates of the inscribed circle: I[11.5; 4.17993517598]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.055549481° = 140°3'20″ = 0.69771631336 rad
∠ B' = β' = 140.055549481° = 140°3'20″ = 0.69771631336 rad
∠ C' = γ' = 79.88990103796° = 79°53'20″ = 1.74772663863 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 = 15 ; ; b = 15 ; ; c = 23 ; ;

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

p = a+b+c = 15+15+23 = 53 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53 }{ 2 } = 26.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.5 * (26.5-15)(26.5-15)(26.5-23) } ; ; T = sqrt{ 12266.19 } = 110.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 110.75 }{ 15 } = 14.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 110.75 }{ 15 } = 14.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 110.75 }{ 23 } = 9.63 ; ;

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{ 15**2-15**2-23**2 }{ 2 * 15 * 23 } ) = 39° 56'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-15**2-23**2 }{ 2 * 15 * 23 } ) = 39° 56'40" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-15**2-15**2 }{ 2 * 15 * 15 } ) = 100° 6'40" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 110.75 }{ 26.5 } = 4.18 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 39° 56'40" } = 11.68 ; ;




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