10 27 30 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 27   c = 30

Area: T = 133.8288014631
Perimeter: p = 67
Semiperimeter: s = 33.5

Angle ∠ A = α = 19.29554560828° = 19°17'44″ = 0.33767692393 rad
Angle ∠ B = β = 63.14993340659° = 63°8'58″ = 1.10221637999 rad
Angle ∠ C = γ = 97.55552098513° = 97°33'19″ = 1.70326596144 rad

Height: ha = 26.76656029261
Height: hb = 9.91331862689
Height: hc = 8.9221867642

Median: ma = 28.09880426365
Median: mb = 17.826554347
Median: mc = 13.76658998979

Inradius: r = 3.99548661084
Circumradius: R = 15.13113609904

Vertex coordinates: A[30; 0] B[0; 0] C[4.51766666667; 8.9221867642]
Centroid: CG[11.50655555556; 2.97439558807]
Coordinates of the circumscribed circle: U[15; -1.98994937598]
Coordinates of the inscribed circle: I[6.5; 3.99548661084]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.7054543917° = 160°42'16″ = 0.33767692393 rad
∠ B' = β' = 116.8510665934° = 116°51'2″ = 1.10221637999 rad
∠ C' = γ' = 82.44547901487° = 82°26'41″ = 1.70326596144 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 = 30 ; ;

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

p = a+b+c = 10+27+30 = 67 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67 }{ 2 } = 33.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.5 * (33.5-10)(33.5-27)(33.5-30) } ; ; T = sqrt{ 17909.94 } = 133.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 133.83 }{ 10 } = 26.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 133.83 }{ 27 } = 9.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 133.83 }{ 30 } = 8.92 ; ;

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{ 10**2-27**2-30**2 }{ 2 * 27 * 30 } ) = 19° 17'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-10**2-30**2 }{ 2 * 10 * 30 } ) = 63° 8'58" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-10**2-27**2 }{ 2 * 27 * 10 } ) = 97° 33'19" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 133.83 }{ 33.5 } = 3.99 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 19° 17'44" } = 15.13 ; ;




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