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 = 0.03   b = 0.05219615242   c = 0.06

Area: T = 0.00107794229
Perimeter: p = 0.14219615242
Semiperimeter: s = 0.07109807621

Angle ∠ A = α = 30° = 0.52435987756 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 0.05219615242
Height: hb = 0.03
Height: hc = 0.02659807621

Median: ma = 0.05440832691
Median: mb = 0.04396862697
Median: mc = 0.03

Inradius: r = 0.01109807621
Circumradius: R = 0.03

Vertex coordinates: A[0.06; 0] B[0; 0] C[0.015; 0.02659807621]
Centroid: CG[0.025; 0.0098660254]
Coordinates of the circumscribed circle: U[0.03; 0]
Coordinates of the inscribed circle: I[0.01990192379; 0.01109807621]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150° = 0.52435987756 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

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

a = 0.03 ; ; c = 0.06 ; ; alpha = 30° ; ;

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 ; ; ; ; 0.03**2 = 0.06**2 + b**2 - 2 * 0.06 * b * cos(30° ) ; ; ; ; ; ; b**2 -0.104b +0.003 =0 ; ; a=1; b=-0.104; c=0.003 ; ; D = b**2 - 4ac = 0.104**2 - 4 * 1 * 0.003 = 0 ; ; D=0 ; ; ; ; b_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 0.1 ± sqrt{ 0 } }{ 2 } ; ; b_{1,2} = fraction{ 0.1 ± 0 }{ 2 } ; ; b_{1,2} = 0.05196152 ± 0 ; ; b_{1} = b_{2} = 0.05196152 ; ; ; ; ; ;
(b -0.05196152) (b -0.05196152) = 0 ; ; ; ; b > 0 ; ; ; ; b = 0.052 ; ;


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 = 0.03 ; ; b = 0.05 ; ; c = 0.06 ; ;

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

p = a+b+c = 0.03+0.05+0.06 = 0.14 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 0.14 }{ 2 } = 0.07 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 0.07 * (0.07-0.03)(0.07-0.05)(0.07-0.06) } ; ; T = sqrt{ 0 } = 0 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0 }{ 0.03 } = 0.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0 }{ 0.05 } = 0.03 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0 }{ 0.06 } = 0.03 ; ;

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{ 0.03**2-0.05**2-0.06**2 }{ 2 * 0.05 * 0.06 } ) = 30° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 0.05**2-0.03**2-0.06**2 }{ 2 * 0.03 * 0.06 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 0.06**2-0.03**2-0.05**2 }{ 2 * 0.05 * 0.03 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0 }{ 0.07 } = 0.01 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 0.03 }{ 2 * sin 30° } = 0.03 ; ;




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