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 = 96   b = 72   c = 120

Area: T = 3456
Perimeter: p = 288
Semiperimeter: s = 144

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 = 72
Height: hb = 96
Height: hc = 57.6

Median: ma = 86.53332306111
Median: mb = 102.5288044944
Median: mc = 60

Inradius: r = 24
Circumradius: R = 60

Vertex coordinates: A[120; 0] B[0; 0] C[76.8; 57.6]
Centroid: CG[65.6; 19.2]
Coordinates of the circumscribed circle: U[60; -0]
Coordinates of the inscribed circle: I[72; 24]

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 hypotenuse c

a = 96 ; ; c = 120 ; ;

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{ 120**2 - 96**2 } = 72 ; ;


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 = 96 ; ; b = 72 ; ; c = 120 ; ;

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

p = a+b+c = 96+72+120 = 288 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 288 }{ 2 } = 144 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 144 * (144-96)(144-72)(144-120) } ; ; T = sqrt{ 11943936 } = 3456 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3456 }{ 96 } = 72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3456 }{ 72 } = 96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3456 }{ 120 } = 57.6 ; ;

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{ 96**2-72**2-120**2 }{ 2 * 72 * 120 } ) = 53° 7'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 72**2-96**2-120**2 }{ 2 * 96 * 120 } ) = 36° 52'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 120**2-96**2-72**2 }{ 2 * 72 * 96 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3456 }{ 144 } = 24 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 96 }{ 2 * sin 53° 7'48" } = 60 ; ;
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