10 27 27 triangle

Acute isosceles triangle.

Sides: a = 10   b = 27   c = 27

Area: T = 132.6654991614
Perimeter: p = 64
Semiperimeter: s = 32

Angle ∠ A = α = 21.34438585715° = 21°20'38″ = 0.37325206072 rad
Angle ∠ B = β = 79.32880707142° = 79°19'41″ = 1.38545360232 rad
Angle ∠ C = γ = 79.32880707142° = 79°19'41″ = 1.38545360232 rad

Height: ha = 26.53329983228
Height: hb = 9.82770364159
Height: hc = 9.82770364159

Median: ma = 26.53329983228
Median: mb = 15.24397506541
Median: mc = 15.24397506541

Inradius: r = 4.14657809879
Circumradius: R = 13.73876106373

Vertex coordinates: A[27; 0] B[0; 0] C[1.85218518519; 9.82770364159]
Centroid: CG[9.61772839506; 3.27656788053]
Coordinates of the circumscribed circle: U[13.5; 2.54440019699]
Coordinates of the inscribed circle: I[5; 4.14657809879]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.6566141428° = 158°39'22″ = 0.37325206072 rad
∠ B' = β' = 100.6721929286° = 100°40'19″ = 1.38545360232 rad
∠ C' = γ' = 100.6721929286° = 100°40'19″ = 1.38545360232 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 = 10 ; ; b = 27 ; ; c = 27 ; ;

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

p = a+b+c = 10+27+27 = 64 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 64 }{ 2 } = 32 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32 * (32-10)(32-27)(32-27) } ; ; T = sqrt{ 17600 } = 132.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 132.66 }{ 10 } = 26.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 132.66 }{ 27 } = 9.83 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 132.66 }{ 27 } = 9.83 ; ;

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{ 27**2+27**2-10**2 }{ 2 * 27 * 27 } ) = 21° 20'38" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10**2+27**2-27**2 }{ 2 * 10 * 27 } ) = 79° 19'41" ; ; gamma = 180° - alpha - beta = 180° - 21° 20'38" - 79° 19'41" = 79° 19'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 132.66 }{ 32 } = 4.15 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10 }{ 2 * sin 21° 20'38" } = 13.74 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 27**2 - 10**2 } }{ 2 } = 26.533 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 10**2 - 27**2 } }{ 2 } = 15.24 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 10**2 - 27**2 } }{ 2 } = 15.24 ; ;
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