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 = 97.35498844375   b = 64.98999229584   c = 117

Area: T = 3159
Perimeter: p = 279.2549807396
Semiperimeter: s = 139.6254903698

Angle ∠ A = α = 56.3109932474° = 56°18'36″ = 0.98327937232 rad
Angle ∠ B = β = 33.6990067526° = 33°41'24″ = 0.58880026035 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 64.98999229584
Height: hb = 97.35498844375
Height: hc = 54

Median: ma = 81.12549036979
Median: mb = 102.6165788259
Median: mc = 58.5

Inradius: r = 22.62549036979
Circumradius: R = 58.5

Vertex coordinates: A[117; 0] B[0; 0] C[81; 54]
Centroid: CG[66; 18]
Coordinates of the circumscribed circle: U[58.5; -0]
Coordinates of the inscribed circle: I[74.72549807396; 22.62549036979]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.6990067526° = 123°41'24″ = 0.98327937232 rad
∠ B' = β' = 146.3109932474° = 146°18'36″ = 0.58880026035 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 = 117 ; ; h = 54 ; ;

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 -117 * c_1 + 2916 = 0 ; ; ; ; c_1 = 81 ; ; c_2 = 36 ; ; ; ; a = sqrt{ c_1**2+h**2 } = sqrt{ 81**2+54**2 } = 97.35 ; ; b = sqrt{ c_2**2+h**2 } = sqrt{ 36**2+54**2 } = 64.9 ; ;


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 = 97.35 ; ; b = 64.9 ; ; c = 117 ; ;

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

p = a+b+c = 97.35+64.9+117 = 279.25 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 279.25 }{ 2 } = 139.62 ; ;

5. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 97.35 * 64.9 }{ 2 } = 3159 ; ;

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

h _a = b = 64.9 ; ; h _b = a = 97.35 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3159 }{ 117 } = 54 ; ;

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{ 97.35 }{ 117 } ) = 56° 18'36" ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 64.9 }{ 117 } ) = 33° 41'24" ; ; gamma = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3159 }{ 139.62 } = 22.62 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 97.35 }{ 2 * sin 56° 18'36" } = 58.5 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 64.9**2+2 * 117**2 - 97.35**2 } }{ 2 } = 81.125 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 117**2+2 * 97.35**2 - 64.9**2 } }{ 2 } = 102.616 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 64.9**2+2 * 97.35**2 - 117**2 } }{ 2 } = 58.5 ; ;
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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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