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 height h

c = 117 ; ; hc = 54 ; ;

2. From hypotenuse c and 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 using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 139.62 * (139.62-97.35)(139.62-64.9)(139.62-117) } ; ; T = sqrt{ 9979281 } = 3159 ; ;

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

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

7. 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 97.35**2-64.9**2-117**2 }{ 2 * 64.9 * 117 } ) = 56° 18'36" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 64.9**2-97.35**2-117**2 }{ 2 * 97.35 * 117 } ) = 33° 41'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 117**2-97.35**2-64.9**2 }{ 2 * 64.9 * 97.35 } ) = 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 ; ;
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