Right triangle calculator (B,c)

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.

Right scalene triangle.

Sides: a = 8.08439459095   b = 3.34884800332   c = 8.75

Area: T = 13.53444657336
Perimeter: p = 20.18224259427
Semiperimeter: s = 10.09112129713

Angle ∠ A = α = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ B = β = 22.5° = 22°30' = 0.39326990817 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 3.34884800332
Height: hb = 8.08439459095
Height: hc = 3.09435921677

Median: ma = 5.24987964239
Median: mb = 8.25554988402
Median: mc = 4.375

Inradius: r = 1.34112129713
Circumradius: R = 4.375

Vertex coordinates: A[8.75; 0] B[0; 0] C[7.46985921677; 3.09435921677]
Centroid: CG[5.40661973892; 1.03111973892]
Coordinates of the circumscribed circle: U[4.375; 0]
Coordinates of the inscribed circle: I[6.74327329381; 1.34112129713]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.5° = 112°30' = 1.17880972451 rad
∠ B' = β' = 157.5° = 157°30' = 0.39326990817 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: hypotenuse c

c = 8.75 ; ;

2. From we calculate angle α:

 alpha + beta + 90° = 180° ; ; alpha = 90° - beta = 90° - 22.5 ° = 67.5 ° ; ;

3. From hypotenuse c and angle α we calculate cathetus a:

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(67.5 ° ) = 8.084 ; ;

4. 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{ 8.75**2 - 8.084**2 } = 3.348 ; ;


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 = 8.08 ; ; b = 3.35 ; ; c = 8.75 ; ;

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

p = a+b+c = 8.08+3.35+8.75 = 20.18 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.18 }{ 2 } = 10.09 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.09 * (10.09-8.08)(10.09-3.35)(10.09-8.75) } ; ; T = sqrt{ 183.18 } = 13.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13.53 }{ 8.08 } = 3.35 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13.53 }{ 3.35 } = 8.08 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13.53 }{ 8.75 } = 3.09 ; ;

9. 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{ 8.08**2-3.35**2-8.75**2 }{ 2 * 3.35 * 8.75 } ) = 67° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.35**2-8.08**2-8.75**2 }{ 2 * 8.08 * 8.75 } ) = 22° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.75**2-8.08**2-3.35**2 }{ 2 * 3.35 * 8.08 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13.53 }{ 10.09 } = 1.34 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.08 }{ 2 * sin 67° 30' } = 4.38 ; ;
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