Right triangle calculator (a,c)

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 hypotenuse c.

Right scalene Pythagorean triangle.

Sides: a = 84   b = 63   c = 105

Area: T = 2646
Perimeter: p = 252
Semiperimeter: s = 126

Angle ∠ A = α = 53.13301023542° = 53°7'48″ = 0.9277295218 rad
Angle ∠ B = β = 36.87698976458° = 36°52'12″ = 0.64435011088 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 63
Height: hb = 84
Height: hc = 50.4

Median: ma = 75.71765767847
Median: mb = 89.71220393258
Median: mc = 52.5

Inradius: r = 21
Circumradius: R = 52.5

Vertex coordinates: A[105; 0] B[0; 0] C[67.2; 50.4]
Centroid: CG[57.4; 16.8]
Coordinates of the circumscribed circle: U[52.5; -0]
Coordinates of the inscribed circle: I[63; 21]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.8769897646° = 126°52'12″ = 0.9277295218 rad
∠ B' = β' = 143.1330102354° = 143°7'48″ = 0.64435011088 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a and hypotenuse c

a = 84 ; ; c = 105 ; ;

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

c**2 = a**2+b**2 ; ; b = sqrt{ c**2 - a**2 } = sqrt{ 105**2 - 84**2 } = 63 ; ;
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 = 84 ; ; b = 63 ; ; c = 105 ; ;

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

p = a+b+c = 84+63+105 = 252 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 252 }{ 2 } = 126 ; ;

5. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 84 * 63 }{ 2 } = 2646 ; ;

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

h _a = b = 63 ; ; h _b = a = 84 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2646 }{ 105 } = 50.4 ; ;

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{ 84 }{ 105 } ) = 53° 7'48" ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 63 }{ 105 } ) = 36° 52'12" ; ; gamma = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2646 }{ 126 } = 21 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 84 }{ 2 * sin 53° 7'48" } = 52.5 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 63**2+2 * 105**2 - 84**2 } }{ 2 } = 75.717 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 105**2+2 * 84**2 - 63**2 } }{ 2 } = 89.712 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 63**2+2 * 84**2 - 105**2 } }{ 2 } = 52.5 ; ;
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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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