Triangle calculator

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

You have entered side a, b and angle α.

Right scalene triangle.

Sides: a = 85   b = 45   c = 72.11110255093

Area: T = 1622.498807396
Perimeter: p = 202.1111025509
Semiperimeter: s = 101.0565512755

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 31.96657187508° = 31°57'57″ = 0.558790704 rad
Angle ∠ C = γ = 58.03442812492° = 58°2'3″ = 1.01328892868 rad

Height: ha = 38.17664252696
Height: hb = 72.11110255093
Height: hc = 45

Median: ma = 42.5
Median: mb = 75.54397246487
Median: mc = 57.66328129734

Inradius: r = 16.05655127546
Circumradius: R = 42.5

Vertex coordinates: A[72.11110255093; 0] B[0; 0] C[72.11110255093; 45]
Centroid: CG[48.07440170062; 15]
Coordinates of the circumscribed circle: U[36.05655127546; 22.5]
Coordinates of the inscribed circle: I[56.05655127546; 16.05655127546]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 148.0344281249° = 148°2'3″ = 0.558790704 rad
∠ C' = γ' = 121.9665718751° = 121°57'57″ = 1.01328892868 rad

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How did we calculate this triangle?

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

a = 85 ; ; b = 45 ; ; alpha = 90° ; ;

2. From angle α, b and side a we calculate c - by using the law of cosines and quadratic equation:

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 85**2 = 45**2 + c**2 - 2 * 45 * c * cos(90° ) ; ; ; ; ; ; c**2 -5200 =0 ; ; a=1; b=-0; c=-5200 ; ; D = b**2 - 4ac = 0**2 - 4 * 1 * (-5200) = 20800 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ ± sqrt{ 20800 } }{ 2 } = fraction{ ± 40 sqrt{ 13 } }{ 2 } ; ; c_{1,2} = ± 20 sqrt{ 13} = ± 72.1110255093 ; ; c_{1} = 20 sqrt{ 13} = 72.1110255093 ; ;
c_{2} = - 20 sqrt{ 13} = -72.1110255093 ; ; ; ; text{ Factored form: } ; ; (c -72.1110255093) (c +72.1110255093) = 0 ; ; ; ; c > 0 ; ; ; ; c = 72.111 ; ;


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 = 85 ; ; b = 45 ; ; c = 72.11 ; ;

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

p = a+b+c = 85+45+72.11 = 202.11 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 202.11 }{ 2 } = 101.06 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 101.06 * (101.06-85)(101.06-45)(101.06-72.11) } ; ; T = sqrt{ 2632500 } = 1622.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1622.5 }{ 85 } = 38.18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1622.5 }{ 45 } = 72.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1622.5 }{ 72.11 } = 45 ; ;

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{ 45**2+72.11**2-85**2 }{ 2 * 45 * 72.11 } ) = 90° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 85**2+72.11**2-45**2 }{ 2 * 85 * 72.11 } ) = 31° 57'57" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 85**2+45**2-72.11**2 }{ 2 * 85 * 45 } ) = 58° 2'3" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1622.5 }{ 101.06 } = 16.06 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 85 }{ 2 * sin 90° } = 42.5 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 72.11**2 - 85**2 } }{ 2 } = 42.5 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 72.11**2+2 * 85**2 - 45**2 } }{ 2 } = 75.54 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 85**2 - 72.11**2 } }{ 2 } = 57.663 ; ;
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