Triangle calculator

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

You have entered side a, c and angle α.

Right scalene triangle.

Sides: a = 111.8   b = 49.99223994223   c = 100

Area: T = 2499.621997112
Perimeter: p = 261.7922399422
Semiperimeter: s = 130.8966199711

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 26.5621567223° = 26°33'42″ = 0.46435868025 rad
Angle ∠ C = γ = 63.4388432777° = 63°26'18″ = 1.10772095243 rad

Height: ha = 44.71659207713
Height: hb = 100
Height: hc = 49.99223994223

Median: ma = 55.9
Median: mb = 103.0776719001
Median: mc = 70.70553039029

Inradius: r = 19.09661997112
Circumradius: R = 55.9

Vertex coordinates: A[100; 0] B[0; 0] C[100; 49.99223994223]
Centroid: CG[66.66766666667; 16.66441331408]
Coordinates of the circumscribed circle: U[50; 24.99661997112]
Coordinates of the inscribed circle: I[80.90438002888; 19.09661997112]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 153.4388432777° = 153°26'18″ = 0.46435868025 rad
∠ C' = γ' = 116.5621567223° = 116°33'42″ = 1.10772095243 rad

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How did we calculate this triangle?

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

a = 111.8 ; ; c = 100 ; ; alpha = 90° ; ;

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 111.8**2 = 100**2 + b**2 - 2 * 100 * b * cos(90° ) ; ; ; ; ; ; b**2 -2499.24 =0 ; ; a=1; b=-0; c=-2499.24 ; ; D = b**2 - 4ac = 0**2 - 4 * 1 * (-2499.24) = 9996.96 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ ± sqrt{ 9996.96 } }{ 2 } ; ; b_{1,2} = ± 49.9923994223 ; ; b_{1} = 49.9923994223 ; ; b_{2} = -49.9923994223 ; ; ; ; (b -49.9923994223) (b +49.9923994223) = 0 ; ;
 ; ; b > 0 ; ; ; ; b = 49.992 ; ;


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 = 111.8 ; ; b = 49.99 ; ; c = 100 ; ;

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

p = a+b+c = 111.8+49.99+100 = 261.79 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 261.79 }{ 2 } = 130.9 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 130.9 * (130.9-111.8)(130.9-49.99)(130.9-100) } ; ; T = sqrt{ 6248100 } = 2499.62 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2499.62 }{ 111.8 } = 44.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2499.62 }{ 49.99 } = 100 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2499.62 }{ 100 } = 49.99 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 111.8**2-49.99**2-100**2 }{ 2 * 49.99 * 100 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 49.99**2-111.8**2-100**2 }{ 2 * 111.8 * 100 } ) = 26° 33'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 100**2-111.8**2-49.99**2 }{ 2 * 49.99 * 111.8 } ) = 63° 26'18" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2499.62 }{ 130.9 } = 19.1 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 111.8 }{ 2 * sin 90° } = 55.9 ; ;




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