Triangle calculator

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

You have entered side a, b and angle β.

Acute scalene triangle.

Sides: a = 9.7   b = 11.8   c = 12.2044005321

Area: T = 53.1999103519
Perimeter: p = 33.7044005321
Semiperimeter: s = 16.85220026605

Angle ∠ A = α = 47.63326246401° = 47°37'57″ = 0.83113461313 rad
Angle ∠ B = β = 64° = 1.11770107213 rad
Angle ∠ C = γ = 68.36773753599° = 68°22'3″ = 1.1933235801 rad

Height: ha = 10.96988873235
Height: hb = 9.01767972066
Height: hc = 8.71883022491

Median: ma = 10.98802719883
Median: mb = 9.31114914454
Median: mc = 8.91223825957

Inradius: r = 3.15768416283
Circumradius: R = 6.56443514488

Vertex coordinates: A[12.2044005321; 0] B[0; 0] C[4.25222001239; 8.71883022491]
Centroid: CG[5.4855401815; 2.90661007497]
Coordinates of the circumscribed circle: U[6.10220026605; 2.42199738583]
Coordinates of the inscribed circle: I[5.05220026605; 3.15768416283]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.367737536° = 132°22'3″ = 0.83113461313 rad
∠ B' = β' = 116° = 1.11770107213 rad
∠ C' = γ' = 111.633262464° = 111°37'57″ = 1.1933235801 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and angle β.

a = 9.7 ; ; b = 11.8 ; ; beta = 64° ; ;

2. From angle β, side a and side b we calculate side c - by using the law of cosines and quadratic equation:

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 11.8**2 = 9.7**2 + c**2 - 2 * 9.7 * c * cos 64° ; ; ; ; ; ; c**2 -8.504c -45.15 =0 ; ; a=1; b=-8.504; c=-45.15 ; ; D = b**2 - 4ac = 8.504**2 - 4 * 1 * (-45.15) = 252.924823573 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 8.5 ± sqrt{ 252.92 } }{ 2 } ; ; c_{1,2} = 4.25220012 ± 7.95180519714 ; ; c_{1} = 12.2040053171 ; ; c_{2} = -3.69960507714 ; ; ; ;
 text{ Factored form: } ; ; (c -12.2040053171) (c +3.69960507714) = 0 ; ; ; ; c > 0 ; ; ; ; c = 12.204 ; ;


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 = 9.7 ; ; b = 11.8 ; ; c = 12.2 ; ;

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

p = a+b+c = 9.7+11.8+12.2 = 33.7 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33.7 }{ 2 } = 16.85 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.85 * (16.85-9.7)(16.85-11.8)(16.85-12.2) } ; ; T = sqrt{ 2830.14 } = 53.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 53.2 }{ 9.7 } = 10.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 53.2 }{ 11.8 } = 9.02 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 53.2 }{ 12.2 } = 8.72 ; ;

7. 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{ 11.8**2+12.2**2-9.7**2 }{ 2 * 11.8 * 12.2 } ) = 47° 37'57" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9.7**2+12.2**2-11.8**2 }{ 2 * 9.7 * 12.2 } ) = 64° ; ; gamma = 180° - alpha - beta = 180° - 47° 37'57" - 64° = 68° 22'3" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 53.2 }{ 16.85 } = 3.16 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 9.7 }{ 2 * sin 47° 37'57" } = 6.56 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.8**2+2 * 12.2**2 - 9.7**2 } }{ 2 } = 10.98 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.2**2+2 * 9.7**2 - 11.8**2 } }{ 2 } = 9.311 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.8**2+2 * 9.7**2 - 12.2**2 } }{ 2 } = 8.912 ; ;
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