Triangle calculator

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

You have entered side c, angle α and angle β.

Right isosceles triangle.

Sides: a = 21.12548150879   b = 21.12548150879   c = 29.875

Area: T = 223.129890625
Perimeter: p = 72.12546301759
Semiperimeter: s = 36.06223150879

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 21.12548150879
Height: hb = 21.12548150879
Height: hc = 14.93875

Median: ma = 23.61882612744
Median: mb = 23.61882612744
Median: mc = 14.93875

Inradius: r = 6.18773150879
Circumradius: R = 14.93875

Vertex coordinates: A[29.875; 0] B[0; 0] C[14.93875; 14.93875]
Centroid: CG[14.93875; 4.97991666667]
Coordinates of the circumscribed circle: U[14.93875; -0]
Coordinates of the inscribed circle: I[14.93875; 6.18773150879]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

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

c = 29.875 ; ; alpha = 45° ; ; beta = 45° ; ;

2. From angle α and angle β we calculate angle γ:

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 45 ° - 45 ° = 90 ° ; ;

3. From angle α, angle γ and side c we calculate side a - By using the law of sines, we calculate unknown side a:

 fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 29.88 * fraction{ sin 45° }{ sin 90° } = 21.12 ; ;

4. 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{ 21.12**2+29.88**2 - 2 * 21.12 * 29.88 * cos(45° ) } ; ; b = 21.12 ; ;


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 = 21.12 ; ; b = 21.12 ; ; c = 29.88 ; ;

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

p = a+b+c = 21.12+21.12+29.88 = 72.12 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72.12 }{ 2 } = 36.06 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36.06 * (36.06-21.12)(36.06-21.12)(36.06-29.88) } ; ; T = sqrt{ 49786.51 } = 223.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 223.13 }{ 21.12 } = 21.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 223.13 }{ 21.12 } = 21.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 223.13 }{ 29.88 } = 14.94 ; ;

9. 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{ 21.12**2+29.88**2-21.12**2 }{ 2 * 21.12 * 29.88 } ) = 45° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 21.12**2+29.88**2-21.12**2 }{ 2 * 21.12 * 29.88 } ) = 45° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 21.12**2+21.12**2-29.88**2 }{ 2 * 21.12 * 21.12 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 223.13 }{ 36.06 } = 6.19 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21.12 }{ 2 * sin 45° } = 14.94 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.12**2+2 * 29.88**2 - 21.12**2 } }{ 2 } = 23.618 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 29.88**2+2 * 21.12**2 - 21.12**2 } }{ 2 } = 23.618 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.12**2+2 * 21.12**2 - 29.88**2 } }{ 2 } = 14.938 ; ;
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