Triangle calculator - result

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, b and c.

Acute scalene triangle.

Sides: a = 8   b = 11   c = 12

Area: T = 42.78994554768
Perimeter: p = 31
Semiperimeter: s = 15.5

Angle ∠ A = α = 40.41554390215° = 40°24'56″ = 0.70553824796 rad
Angle ∠ B = β = 63.05656418195° = 63°3'20″ = 1.10105285617 rad
Angle ∠ C = γ = 76.5298919159° = 76°31'44″ = 1.33656816123 rad

Height: ha = 10.69773638692
Height: hb = 7.78799009958
Height: hc = 7.13215759128

Median: ma = 10.79435165725
Median: mb = 8.58877820187
Median: mc = 7.51766481892

Inradius: r = 2.76106100308
Circumradius: R = 6.17697443227

Vertex coordinates: A[12; 0] B[0; 0] C[3.625; 7.13215759128]
Centroid: CG[5.20883333333; 2.37771919709]
Coordinates of the circumscribed circle: U[6; 1.43772699843]
Coordinates of the inscribed circle: I[4.5; 2.76106100308]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.5854560979° = 139°35'4″ = 0.70553824796 rad
∠ B' = β' = 116.944435818° = 116°56'40″ = 1.10105285617 rad
∠ C' = γ' = 103.4711080841° = 103°28'16″ = 1.33656816123 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and c.

a = 8 ; ; b = 11 ; ; c = 12 ; ;


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 = 8 ; ; b = 11 ; ; c = 12 ; ; : Nr. 1

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

p = a+b+c = 8+11+12 = 31 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31 }{ 2 } = 15.5 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.5 * (15.5-8)(15.5-11)(15.5-12) } ; ; T = sqrt{ 1830.94 } = 42.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 42.79 }{ 8 } = 10.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 42.79 }{ 11 } = 7.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 42.79 }{ 12 } = 7.13 ; ;

6. 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{ 11**2+12**2-8**2 }{ 2 * 11 * 12 } ) = 40° 24'56" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8**2+12**2-11**2 }{ 2 * 8 * 12 } ) = 63° 3'20" ; ; gamma = 180° - alpha - beta = 180° - 40° 24'56" - 63° 3'20" = 76° 31'44" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 42.79 }{ 15.5 } = 2.76 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8 }{ 2 * sin 40° 24'56" } = 6.17 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 11**2+2 * 12**2 - 8**2 } }{ 2 } = 10.794 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 8**2 - 11**2 } }{ 2 } = 8.588 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 11**2+2 * 8**2 - 12**2 } }{ 2 } = 7.517 ; ;
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