Right triangle calculator (a,b)

Please enter two properties of the right triangle

Use symbols: a, b, c, A, B, h, T, p, r, R


You have entered cathetus a and cathetus b.

Right scalene triangle.

Sides: a = 110   b = 420   c = 434.1665866922

Area: T = 23100
Perimeter: p = 964.1665866922
Semiperimeter: s = 482.0832933461

Angle ∠ A = α = 14.67663931375° = 14°40'35″ = 0.25661513826 rad
Angle ∠ B = β = 75.32436068625° = 75°19'25″ = 1.31546449442 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 420
Height: hb = 110
Height: hc = 106.4110944572

Median: ma = 423.5865882673
Median: mb = 237.0655391823
Median: mc = 217.0832933461

Inradius: r = 47.91770665391
Circumradius: R = 217.0832933461

Vertex coordinates: A[434.1665866922; 0] B[0; 0] C[27.87695331021; 106.4110944572]
Centroid: CG[154.0121800008; 35.47703148573]
Coordinates of the circumscribed circle: U[217.0832933461; -0]
Coordinates of the inscribed circle: I[62.08329334609; 47.91770665391]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.3243606863° = 165°19'25″ = 0.25661513826 rad
∠ B' = β' = 104.6766393137° = 104°40'35″ = 1.31546449442 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a and cathetus b

a = 110 ; ; b = 420 ; ;

2. From cathetus a and cathetus b we calculate hypotenuse c - Pythagorean theorem:

c**2 = a**2+b**2 ; ; c = sqrt{ a**2+b**2 } = sqrt{ 110**2 + 420**2 } = 434.166 ; ;


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 = 420 ; ; c = 434.17 ; ;

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

p = a+b+c = 110+420+434.17 = 964.17 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 964.17 }{ 2 } = 482.08 ; ;

5. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 110 * 420 }{ 2 } = 23100 ; ;

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

h _a = b = 420 ; ; h _b = a = 110 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 23100 }{ 434.17 } = 106.41 ; ;

7. Calculation of the inner angles of the triangle - basic use of sine function

 sin alpha = fraction{ a }{ c } ; ; alpha = arcsin( fraction{ a }{ c } ) = arcsin( fraction{ 110 }{ 434.17 } ) = 14° 40'35" ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 420 }{ 434.17 } ) = 75° 19'25" ; ; gamma = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 23100 }{ 482.08 } = 47.92 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 110 }{ 2 * sin 14° 40'35" } = 217.08 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 420**2+2 * 434.17**2 - 110**2 } }{ 2 } = 423.586 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 434.17**2+2 * 110**2 - 420**2 } }{ 2 } = 237.065 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 420**2+2 * 110**2 - 434.17**2 } }{ 2 } = 217.083 ; ;
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Trigonometry right triangle solver. You can calculate angles, sides (adjacent, opposite, hypotenuse) and area of any right-angled triangle and use it in real world to find height. A right-angled triangle is entirely determined by two independent properties. Step-by-step explanations are provided for each calculation.

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