Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, b and angle γ.

Right scalene triangle.

Sides: a = 12.7   b = 72   c = 73.1111490205

Area: T = 457.2
Perimeter: p = 157.8111490205
Semiperimeter: s = 78.90657451025

Angle ∠ A = α = 10.00334400836° = 10°12″ = 0.1754592966 rad
Angle ∠ B = β = 79.99765599164° = 79°59'48″ = 1.39662033608 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 72
Height: hb = 12.7
Height: hc = 12.50769260309

Median: ma = 72.27994749566
Median: mb = 38.17444679072
Median: mc = 36.55657451025

Inradius: r = 5.79442548975
Circumradius: R = 36.55657451025

Vertex coordinates: A[73.1111490205; 0] B[0; 0] C[2.2066082786; 12.50769260309]
Centroid: CG[25.10658576637; 4.16989753436]
Coordinates of the circumscribed circle: U[36.55657451025; 0]
Coordinates of the inscribed circle: I[6.90657451025; 5.79442548975]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 169.9976559916° = 169°59'48″ = 0.1754592966 rad
∠ B' = β' = 100.0033440084° = 100°12″ = 1.39662033608 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and angle γ.

a = 12.7 ; ; b = 72 ; ; gamma = 90° ; ;

2. Calculation of the third side c of the triangle using a Law of Cosines

c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 12.7**2+72**2 - 2 * 12.7 * 72 * cos 90° } ; ; c = 73.11 ; ;


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 = 12.7 ; ; b = 72 ; ; c = 73.11 ; ;

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

p = a+b+c = 12.7+72+73.11 = 157.81 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 157.81 }{ 2 } = 78.91 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 78.91 * (78.91-12.7)(78.91-72)(78.91-73.11) } ; ; T = sqrt{ 209031.84 } = 457.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 457.2 }{ 12.7 } = 72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 457.2 }{ 72 } = 12.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 457.2 }{ 73.11 } = 12.51 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 72**2+73.11**2-12.7**2 }{ 2 * 72 * 73.11 } ) = 10° 12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.7**2+73.11**2-72**2 }{ 2 * 12.7 * 73.11 } ) = 79° 59'48" ; ; gamma = 180° - alpha - beta = 180° - 10° 12" - 79° 59'48" = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 457.2 }{ 78.91 } = 5.79 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12.7 }{ 2 * sin 10° 12" } = 36.56 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 72**2+2 * 73.11**2 - 12.7**2 } }{ 2 } = 72.279 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 73.11**2+2 * 12.7**2 - 72**2 } }{ 2 } = 38.174 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 72**2+2 * 12.7**2 - 73.11**2 } }{ 2 } = 36.556 ; ;
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