Triangle calculator

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

You have entered height ha, angle α and angle β.

Right isosceles triangle.

Sides: a = 110   b = 77.78217459305   c = 77.78217459305

Area: T = 3025
Perimeter: p = 265.5633491861
Semiperimeter: s = 132.782174593

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

Height: ha = 55
Height: hb = 77.78217459305
Height: hc = 77.78217459305

Median: ma = 55
Median: mb = 86.96326356546
Median: mc = 86.96326356546

Inradius: r = 22.78217459305
Circumradius: R = 55

Vertex coordinates: A[77.78217459305; 0] B[0; 0] C[77.78217459305; 77.78217459305]
Centroid: CG[51.8544497287; 25.92772486435]
Coordinates of the circumscribed circle: U[38.89108729653; 38.89108729653]
Coordinates of the inscribed circle: I[55; 22.78217459305]

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

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How did we calculate this triangle?

1. Input data entered: angle α, angle β and height ha.

alpha = 90° ; ; beta = 45° ; ; h_a = 55 ; ;

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

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

3. From side c and angle α we calculate height hb:

h_b = c * sin alpha = 77.782 * sin 90° = 77.782 ; ;

4. Calculation of the third side a of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 77.78**2+77.78**2 - 2 * 77.78 * 77.78 * cos 90° } ; ; a = 110 ; ;


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 = 110 ; ; b = 77.78 ; ; c = 77.78 ; ;

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

p = a+b+c = 110+77.78+77.78 = 265.56 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 265.56 }{ 2 } = 132.78 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 132.78 * (132.78-110)(132.78-77.78)(132.78-77.78) } ; ; T = sqrt{ 9150625 } = 3025 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3025 }{ 110 } = 55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3025 }{ 77.78 } = 77.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3025 }{ 77.78 } = 77.78 ; ;

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{ 77.78**2+77.78**2-110**2 }{ 2 * 77.78 * 77.78 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 110**2+77.78**2-77.78**2 }{ 2 * 110 * 77.78 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 90° - 45° = 45° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3025 }{ 132.78 } = 22.78 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 110 }{ 2 * sin 90° } = 55 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 77.78**2+2 * 77.78**2 - 110**2 } }{ 2 } = 55 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 77.78**2+2 * 110**2 - 77.78**2 } }{ 2 } = 86.963 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 77.78**2+2 * 110**2 - 77.78**2 } }{ 2 } = 86.963 ; ;
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