Right triangle calculator (c,v)

Please enter two properties of the right triangle

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


You have entered hypotenuse c and height h.

Right scalene triangle.

Sides: a = 145.7743797371   b = 87.46442784227   c = 170

Area: T = 6375
Perimeter: p = 403.2388075794
Semiperimeter: s = 201.6199037897

Angle ∠ A = α = 59.03662434679° = 59°2'10″ = 1.03303768265 rad
Angle ∠ B = β = 30.96437565321° = 30°57'50″ = 0.54404195003 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 87.46442784227
Height: hb = 145.7743797371
Height: hc = 75

Median: ma = 113.8532975367
Median: mb = 152.1922312552
Median: mc = 85

Inradius: r = 31.61990378969
Circumradius: R = 85

Vertex coordinates: A[170; 0] B[0; 0] C[125; 75]
Centroid: CG[98.33333333333; 25]
Coordinates of the circumscribed circle: U[85; -0]
Coordinates of the inscribed circle: I[114.1554759474; 31.61990378969]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120.9643756532° = 120°57'50″ = 1.03303768265 rad
∠ B' = β' = 149.0366243468° = 149°2'10″ = 0.54404195003 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: hypotenuse c and height h

c = 170 ; ; h = 75 ; ;

2. From hypotenuse c and height h we calculate a,b - Pythagorean theorem, Euclid's theorem:

c = c_1+c_2 ; ; h**2 = c_1 * c_2 ; ; ; ; h**2 = c_1 * (c-c_1) ; ; h**2 = c_1 * c-c_1 **2 ; ; ; ; c_1**2 -c_1 * c + h**2 = 0 ; ; ; ; c_1**2 -170 * c_1 + 5625 = 0 ; ; ; ; c_1 = 125 ; ; c_2 = 45 ; ; ; ; a = sqrt{ c_1**2+h**2 } = sqrt{ 125**2+75**2 } = 145.774 ; ; b = sqrt{ c_2**2+h**2 } = sqrt{ 45**2+75**2 } = 87.464 ; ;
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 = 145.77 ; ; b = 87.46 ; ; c = 170 ; ;

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

p = a+b+c = 145.77+87.46+170 = 403.24 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 403.24 }{ 2 } = 201.62 ; ;

5. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 145.77 * 87.46 }{ 2 } = 6375 ; ;

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

h _a = b = 87.46 ; ; h _b = a = 145.77 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6375 }{ 170 } = 75 ; ;

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{ 145.77 }{ 170 } ) = 59° 2'10" ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 87.46 }{ 170 } ) = 30° 57'50" ; ; gamma = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6375 }{ 201.62 } = 31.62 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 145.77 }{ 2 * sin 59° 2'10" } = 85 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 87.46**2+2 * 170**2 - 145.77**2 } }{ 2 } = 113.853 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 170**2+2 * 145.77**2 - 87.46**2 } }{ 2 } = 152.192 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 87.46**2+2 * 145.77**2 - 170**2 } }{ 2 } = 85 ; ;
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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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