Triangle calculator

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

You have entered side a, c and angle β.

Acute scalene triangle.

Sides: a = 10.1   b = 13.76999844563   c = 9.8

Area: T = 49.4222175675
Perimeter: p = 33.65999844563
Semiperimeter: s = 16.87999922281

Angle ∠ A = α = 47.41101317895° = 47°24'36″ = 0.82774628985 rad
Angle ∠ B = β = 87° = 1.51884364492 rad
Angle ∠ C = γ = 45.59898682105° = 45°35'24″ = 0.79656933058 rad

Height: ha = 9.78765694406
Height: hb = 7.21549243428
Height: hc = 10.0866158301

Median: ma = 10.78771352569
Median: mb = 7.21882135238
Median: mc = 10.9932715181

Inradius: r = 2.9421797532
Circumradius: R = 6.85993927876

Vertex coordinates: A[9.8; 0] B[0; 0] C[0.52985931581; 10.0866158301]
Centroid: CG[3.4432864386; 3.3622052767]
Coordinates of the circumscribed circle: U[4.9; 4.88001322288]
Coordinates of the inscribed circle: I[3.11000077719; 2.9421797532]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.598986821° = 132°35'24″ = 0.82774628985 rad
∠ B' = β' = 93° = 1.51884364492 rad
∠ C' = γ' = 134.411013179° = 134°24'36″ = 0.79656933058 rad

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How did we calculate this triangle?

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

a = 10.1 ; ; c = 9.8 ; ; beta = 87° ; ;

2. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 10.1**2+9.8**2 - 2 * 10.1 * 9.8 * cos 87° } ; ; b = 13.7 ; ;


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 = 10.1 ; ; b = 13.7 ; ; c = 9.8 ; ;

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

p = a+b+c = 10.1+13.7+9.8 = 33.6 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33.6 }{ 2 } = 16.8 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.8 * (16.8-10.1)(16.8-13.7)(16.8-9.8) } ; ; T = sqrt{ 2442.55 } = 49.42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 49.42 }{ 10.1 } = 9.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 49.42 }{ 13.7 } = 7.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 49.42 }{ 9.8 } = 10.09 ; ;

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{ 13.7**2+9.8**2-10.1**2 }{ 2 * 13.7 * 9.8 } ) = 47° 24'36" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.1**2+9.8**2-13.7**2 }{ 2 * 10.1 * 9.8 } ) = 87° ; ; gamma = 180° - alpha - beta = 180° - 47° 24'36" - 87° = 45° 35'24" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 49.42 }{ 16.8 } = 2.94 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10.1 }{ 2 * sin 47° 24'36" } = 6.86 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.7**2+2 * 9.8**2 - 10.1**2 } }{ 2 } = 10.787 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.8**2+2 * 10.1**2 - 13.7**2 } }{ 2 } = 7.218 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.7**2+2 * 10.1**2 - 9.8**2 } }{ 2 } = 10.993 ; ;
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