Triangle calculator

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

You have entered side a, b and angle γ.

Right isosceles triangle.

Sides: a = 48   b = 48   c = 67.88222509939

Area: T = 1152
Perimeter: p = 163.8822250994
Semiperimeter: s = 81.9411125497

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

Height: ha = 48
Height: hb = 48
Height: hc = 33.9411125497

Median: ma = 53.666563146
Median: mb = 53.666563146
Median: mc = 33.9411125497

Inradius: r = 14.0598874503
Circumradius: R = 33.9411125497

Vertex coordinates: A[67.88222509939; 0] B[0; 0] C[33.9411125497; 33.9411125497]
Centroid: CG[33.9411125497; 11.3143708499]
Coordinates of the circumscribed circle: U[33.9411125497; 0]
Coordinates of the inscribed circle: I[33.9411125497; 14.0598874503]

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 a, b and angle γ.

a = 48 ; ; b = 48 ; ; gamma = 90° ; ;

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

c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 48**2+48**2 - 2 * 48 * 48 * cos 90° } ; ; c = 67.88 ; ;
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 = 48 ; ; b = 48 ; ; c = 67.88 ; ;

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

p = a+b+c = 48+48+67.88 = 163.88 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 163.88 }{ 2 } = 81.94 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 81.94 * (81.94-48)(81.94-48)(81.94-67.88) } ; ; T = sqrt{ 1327104 } = 1152 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1152 }{ 48 } = 48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1152 }{ 48 } = 48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1152 }{ 67.88 } = 33.94 ; ;

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{ 48**2+67.88**2-48**2 }{ 2 * 48 * 67.88 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 48**2+67.88**2-48**2 }{ 2 * 48 * 67.88 } ) = 45° ; ;
gamma = 180° - alpha - beta = 180° - 45° - 45° = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1152 }{ 81.94 } = 14.06 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 48 }{ 2 * sin 45° } = 33.94 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 67.88**2 - 48**2 } }{ 2 } = 53.666 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 67.88**2+2 * 48**2 - 48**2 } }{ 2 } = 53.666 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 48**2 - 67.88**2 } }{ 2 } = 33.941 ; ;
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