13 13 16 triangle

Acute isosceles triangle.

Sides: a = 13   b = 13   c = 16

Area: T = 81.97656061277
Perimeter: p = 42
Semiperimeter: s = 21

Angle ∠ A = α = 52.02201275551° = 52°1'12″ = 0.90879225031 rad
Angle ∠ B = β = 52.02201275551° = 52°1'12″ = 0.90879225031 rad
Angle ∠ C = γ = 75.96597448897° = 75°57'35″ = 1.32657476473 rad

Height: ha = 12.6121631712
Height: hb = 12.6121631712
Height: hc = 10.2476950766

Median: ma = 13.04879883507
Median: mb = 13.04879883507
Median: mc = 10.2476950766

Inradius: r = 3.90436002918
Circumradius: R = 8.24663556164

Vertex coordinates: A[16; 0] B[0; 0] C[8; 10.2476950766]
Centroid: CG[8; 3.41656502553]
Coordinates of the circumscribed circle: U[8; 2.00105951495]
Coordinates of the inscribed circle: I[8; 3.90436002918]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.9879872445° = 127°58'48″ = 0.90879225031 rad
∠ B' = β' = 127.9879872445° = 127°58'48″ = 0.90879225031 rad
∠ C' = γ' = 104.044025511° = 104°2'25″ = 1.32657476473 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 = 13 ; ; b = 13 ; ; c = 16 ; ;

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

p = a+b+c = 13+13+16 = 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 * (21-13)(21-13)(21-16) } ; ; T = sqrt{ 6720 } = 81.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 81.98 }{ 13 } = 12.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 81.98 }{ 13 } = 12.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 81.98 }{ 16 } = 10.25 ; ;

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{ 13**2-13**2-16**2 }{ 2 * 13 * 16 } ) = 52° 1'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-13**2-16**2 }{ 2 * 13 * 16 } ) = 52° 1'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16**2-13**2-13**2 }{ 2 * 13 * 13 } ) = 75° 57'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 81.98 }{ 21 } = 3.9 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 52° 1'12" } = 8.25 ; ;




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