Triangle calculator

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

You have entered side a, c and angle α.

Triangle has two solutions: a=2.9; b=4.01219596489; c=6.2 and a=2.9; b=7.48551201477; c=6.2.

#1 Obtuse scalene triangle.

Sides: a = 2.9   b = 4.01219596489   c = 6.2

Area: T = 4.65990102647
Perimeter: p = 13.11219596489
Semiperimeter: s = 6.55659798245

Angle ∠ A = α = 22° = 0.38439724354 rad
Angle ∠ B = β = 31.21545114756° = 31°12'52″ = 0.54547959997 rad
Angle ∠ C = γ = 126.7855488524° = 126°47'8″ = 2.21328242185 rad

Height: ha = 3.21331105274
Height: hb = 2.32325608792
Height: hc = 1.5032906537

Median: ma = 5.01765137409
Median: mb = 4.40546617286
Median: mc = 1.62657029595

Inradius: r = 0.71106504885
Circumradius: R = 3.87107273857

Vertex coordinates: A[6.2; 0] B[0; 0] C[2.48801757883; 1.5032906537]
Centroid: CG[2.89333919294; 0.50109688457]
Coordinates of the circumscribed circle: U[3.1; -2.31878719754]
Coordinates of the inscribed circle: I[2.54440201755; 0.71106504885]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158° = 0.38439724354 rad
∠ B' = β' = 148.7855488524° = 148°47'8″ = 0.54547959997 rad
∠ C' = γ' = 53.21545114756° = 53°12'52″ = 2.21328242185 rad




How did we calculate this triangle?

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

a = 2.9 ; ; c = 6.2 ; ; alpha = 22° ; ;

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 2.9**2 = 6.2**2 + b**2 - 2 * 6.2 * b * cos(22° ) ; ; ; ; ; ; b**2 -11.497b +30.03 =0 ; ; a=1; b=-11.497; c=30.03 ; ; D = b**2 - 4ac = 11.497**2 - 4 * 1 * 30.03 = 12.06284385 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 11.5 ± sqrt{ 12.06 } }{ 2 } ; ; b_{1,2} = 5.7485399 ± 1.73658024937 ; ; b_{1} = 7.48512014937 ; ; b_{2} = 4.01195965063 ; ; ; ;
 text{ Factored form: } ; ; (b -7.48512014937) (b -4.01195965063) = 0 ; ; ; ; b > 0 ; ; ; ; b = 7.485 ; ;


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 = 2.9 ; ; b = 4.01 ; ; c = 6.2 ; ;

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

p = a+b+c = 2.9+4.01+6.2 = 13.11 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 13.11 }{ 2 } = 6.56 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.56 * (6.56-2.9)(6.56-4.01)(6.56-6.2) } ; ; T = sqrt{ 21.71 } = 4.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.66 }{ 2.9 } = 3.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.66 }{ 4.01 } = 2.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.66 }{ 6.2 } = 1.5 ; ;

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{ 4.01**2+6.2**2-2.9**2 }{ 2 * 4.01 * 6.2 } ) = 22° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.9**2+6.2**2-4.01**2 }{ 2 * 2.9 * 6.2 } ) = 31° 12'52" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 2.9**2+4.01**2-6.2**2 }{ 2 * 2.9 * 4.01 } ) = 126° 47'8" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.66 }{ 6.56 } = 0.71 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2.9 }{ 2 * sin 22° } = 3.87 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.01**2+2 * 6.2**2 - 2.9**2 } }{ 2 } = 5.017 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.2**2+2 * 2.9**2 - 4.01**2 } }{ 2 } = 4.405 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.01**2+2 * 2.9**2 - 6.2**2 } }{ 2 } = 1.626 ; ;







#2 Obtuse scalene triangle.

Sides: a = 2.9   b = 7.48551201477   c = 6.2

Area: T = 8.69223236155
Perimeter: p = 16.58551201477
Semiperimeter: s = 8.29325600738

Angle ∠ A = α = 22° = 0.38439724354 rad
Angle ∠ B = β = 104.7855488524° = 104°47'8″ = 1.82988517831 rad
Angle ∠ C = γ = 53.21545114756° = 53°12'52″ = 0.92987684351 rad

Height: ha = 5.99547059417
Height: hb = 2.32325608792
Height: hc = 2.80439753598

Median: ma = 6.71879618794
Median: mb = 3.06989157847
Median: mc = 4.75548408819

Inradius: r = 1.04882074942
Circumradius: R = 3.87107273857

Vertex coordinates: A[6.2; 0] B[0; 0] C[-0.74400825504; 2.80439753598]
Centroid: CG[1.82199724832; 0.93546584533]
Coordinates of the circumscribed circle: U[3.1; 2.31878719754]
Coordinates of the inscribed circle: I[0.80774399262; 1.04882074942]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158° = 0.38439724354 rad
∠ B' = β' = 75.21545114756° = 75°12'52″ = 1.82988517831 rad
∠ C' = γ' = 126.7855488524° = 126°47'8″ = 0.92987684351 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 2.9 ; ; c = 6.2 ; ; alpha = 22° ; ; : Nr. 1

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 2.9**2 = 6.2**2 + b**2 - 2 * 6.2 * b * cos(22° ) ; ; ; ; ; ; b**2 -11.497b +30.03 =0 ; ; a=1; b=-11.497; c=30.03 ; ; D = b**2 - 4ac = 11.497**2 - 4 * 1 * 30.03 = 12.06284385 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 11.5 ± sqrt{ 12.06 } }{ 2 } ; ; b_{1,2} = 5.7485399 ± 1.73658024937 ; ; b_{1} = 7.48512014937 ; ; b_{2} = 4.01195965063 ; ; ; ; : Nr. 1
 text{ Factored form: } ; ; (b -7.48512014937) (b -4.01195965063) = 0 ; ; ; ; b > 0 ; ; ; ; b = 7.485 ; ; : Nr. 1


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 = 2.9 ; ; b = 7.49 ; ; c = 6.2 ; ;

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

p = a+b+c = 2.9+7.49+6.2 = 16.59 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 16.59 }{ 2 } = 8.29 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.29 * (8.29-2.9)(8.29-7.49)(8.29-6.2) } ; ; T = sqrt{ 75.56 } = 8.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8.69 }{ 2.9 } = 5.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8.69 }{ 7.49 } = 2.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8.69 }{ 6.2 } = 2.8 ; ;

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{ 7.49**2+6.2**2-2.9**2 }{ 2 * 7.49 * 6.2 } ) = 22° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.9**2+6.2**2-7.49**2 }{ 2 * 2.9 * 6.2 } ) = 104° 47'8" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 2.9**2+7.49**2-6.2**2 }{ 2 * 2.9 * 7.49 } ) = 53° 12'52" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8.69 }{ 8.29 } = 1.05 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2.9 }{ 2 * sin 22° } = 3.87 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.49**2+2 * 6.2**2 - 2.9**2 } }{ 2 } = 6.718 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.2**2+2 * 2.9**2 - 7.49**2 } }{ 2 } = 3.069 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.49**2+2 * 2.9**2 - 6.2**2 } }{ 2 } = 4.755 ; ;
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