27 27 30 triangle

Acute isosceles triangle.

Sides: a = 27   b = 27   c = 30

Area: T = 336.749916481
Perimeter: p = 84
Semiperimeter: s = 42

Angle ∠ A = α = 56.25110114041° = 56°15'4″ = 0.98217653566 rad
Angle ∠ B = β = 56.25110114041° = 56°15'4″ = 0.98217653566 rad
Angle ∠ C = γ = 67.49879771918° = 67°29'53″ = 1.17880619404 rad

Height: ha = 24.94443825785
Height: hb = 24.94443825785
Height: hc = 22.45499443206

Median: ma = 25.14545819214
Median: mb = 25.14545819214
Median: mc = 22.45499443206

Inradius: r = 8.01878372574
Circumradius: R = 16.23661204462

Vertex coordinates: A[30; 0] B[0; 0] C[15; 22.45499443206]
Centroid: CG[15; 7.48333147735]
Coordinates of the circumscribed circle: U[15; 6.21438238745]
Coordinates of the inscribed circle: I[15; 8.01878372574]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.7498988596° = 123°44'56″ = 0.98217653566 rad
∠ B' = β' = 123.7498988596° = 123°44'56″ = 0.98217653566 rad
∠ C' = γ' = 112.5022022808° = 112°30'7″ = 1.17880619404 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 = 27 ; ; b = 27 ; ; c = 30 ; ;

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

p = a+b+c = 27+27+30 = 84 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 84 }{ 2 } = 42 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 42 * (42-27)(42-27)(42-30) } ; ; T = sqrt{ 113400 } = 336.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 336.75 }{ 27 } = 24.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 336.75 }{ 27 } = 24.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 336.75 }{ 30 } = 22.45 ; ;

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+30**2-27**2 }{ 2 * 27 * 30 } ) = 56° 15'4" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 27**2+30**2-27**2 }{ 2 * 27 * 30 } ) = 56° 15'4" ; ; gamma = 180° - alpha - beta = 180° - 56° 15'4" - 56° 15'4" = 67° 29'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 336.75 }{ 42 } = 8.02 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 27 }{ 2 * sin 56° 15'4" } = 16.24 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 30**2 - 27**2 } }{ 2 } = 25.145 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 30**2+2 * 27**2 - 27**2 } }{ 2 } = 25.145 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 27**2 - 30**2 } }{ 2 } = 22.45 ; ;
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