Triangle calculator - result

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

You have entered side a, median mc and angle β.

Triangle has two solutions: a=7; b=4.61096016282; c=6.36437138795 and a=7; b=10.71438515788; c=15.08655305278.

#1 Acute scalene triangle.

Sides: a = 7   b = 4.61096016282   c = 6.36437138795

Area: T = 14.31768075168
Perimeter: p = 17.97333155077
Semiperimeter: s = 8.98766577539

Angle ∠ A = α = 77.45328590336° = 77°27'10″ = 1.35218074052 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 62.54771409664° = 62°32'50″ = 1.09216535476 rad

Height: ha = 4.09105164334
Height: hb = 6.21217331048
Height: hc = 4.54995132678

Median: ma = 4.3155395782
Median: mb = 6.28798344228
Median: mc = 5

Inradius: r = 1.59331181435
Circumradius: R = 3.58656335426

Vertex coordinates: A[6.36437138795; 0] B[0; 0] C[5.36223111018; 4.54995132678]
Centroid: CG[3.90986749938; 1.54998377559]
Coordinates of the circumscribed circle: U[3.18218569398; 1.65330439549]
Coordinates of the inscribed circle: I[4.37770561257; 1.59331181435]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 102.5477140966° = 102°32'50″ = 1.35218074052 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 117.4532859034° = 117°27'10″ = 1.09216535476 rad




How did we calculate this triangle?

1. Input data entered: side a, angle β and median mc.

a = 7 ; ; beta = 40° ; ; m_c = 5 ; ;

2. From side a and angle β we calculate height hc:

h_c = a * sin beta = 7 * sin 40° = 4.5 ; ;

3. From side a, angle β and median mc we calculate side c:

x = c/2 ; ; x**2 - 2 a cos beta + a**2 - m_c**2 = 0 ; ; x**2 -10.725x +24 =0 ; ; a=1; b=-10.725; c=24 ; ; D = b**2 - 4ac = 10.725**2 - 4 * 1 * 24 = 19.0175214114 ; ; D>0 ; ; ; ; x_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 10.72 ± sqrt{ 19.02 } }{ 2 } ; ; x_{1,2} = 5.3623111 ± 2.18045416206 ; ; x_{1} = 7.54276526206 ; ; x_{2} = 3.18185693794 ; ; ; ; text{ Factored form: } ; ; (x -7.54276526206) (x -3.18185693794) = 0 ; ; ; ;
c_1 = 2 * x_1 = 15.086 ; ; c_2 = 2 * x_2 = 6.364 ; ;

4. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 7**2+15.09**2 - 2 * 7 * 15.09 * cos 40° } ; ; b = 10.71 ; ;


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 = 7 ; ; b = 4.61 ; ; c = 6.36 ; ;

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

p = a+b+c = 7+4.61+6.36 = 17.97 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 17.97 }{ 2 } = 8.99 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.99 * (8.99-7)(8.99-4.61)(8.99-6.36) } ; ; T = sqrt{ 204.97 } = 14.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14.32 }{ 7 } = 4.09 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14.32 }{ 4.61 } = 6.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14.32 }{ 6.36 } = 4.5 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 4.61**2+6.36**2-7**2 }{ 2 * 4.61 * 6.36 } ) = 77° 27'10" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7**2+6.36**2-4.61**2 }{ 2 * 7 * 6.36 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 77° 27'10" - 40° = 62° 32'50" ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14.32 }{ 8.99 } = 1.59 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7 }{ 2 * sin 77° 27'10" } = 3.59 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.61**2+2 * 6.36**2 - 7**2 } }{ 2 } = 4.315 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.36**2+2 * 7**2 - 4.61**2 } }{ 2 } = 6.28 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.61**2+2 * 7**2 - 6.36**2 } }{ 2 } = 5 ; ;







#2 Obtuse scalene triangle.

Sides: a = 7   b = 10.71438515788   c = 15.08655305278

Area: T = 33.93987723808
Perimeter: p = 32.79993821066
Semiperimeter: s = 16.43996910533

Angle ∠ A = α = 24.8332793774° = 24°49'58″ = 0.43334140138 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 115.1677206226° = 115°10'2″ = 2.0110046939 rad

Height: ha = 9.69767921088
Height: hb = 6.3355494221
Height: hc = 4.54995132678

Median: ma = 12.60767411919
Median: mb = 10.46985224239
Median: mc = 5

Inradius: r = 2.06994763255
Circumradius: R = 8.33438970893

Vertex coordinates: A[15.08655305278; 0] B[0; 0] C[5.36223111018; 4.54995132678]
Centroid: CG[6.81659472099; 1.54998377559]
Coordinates of the circumscribed circle: U[7.54327652639; -3.54440842073]
Coordinates of the inscribed circle: I[5.68658394745; 2.06994763255]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.1677206226° = 155°10'2″ = 0.43334140138 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 64.8332793774° = 64°49'58″ = 2.0110046939 rad

Calculate another triangle

How did we calculate this triangle?

1. Input data entered: side a, angle β and median mc.

a = 7 ; ; beta = 40° ; ; m_c = 5 ; ; : Nr. 1

2. From side a and angle β we calculate height hc:

h_c = a * sin beta = 7 * sin 40° = 4.5 ; ; : Nr. 1

3. From side a, angle β and median mc we calculate side c:

x = c/2 ; ; x**2 - 2 a cos beta + a**2 - m_c**2 = 0 ; ; x**2 -10.725x +24 =0 ; ; a=1; b=-10.725; c=24 ; ; D = b**2 - 4ac = 10.725**2 - 4 * 1 * 24 = 19.0175214114 ; ; D>0 ; ; ; ; x_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 10.72 ± sqrt{ 19.02 } }{ 2 } ; ; x_{1,2} = 5.3623111 ± 2.18045416206 ; ; x_{1} = 7.54276526206 ; ; x_{2} = 3.18185693794 ; ; ; ; text{ Factored form: } ; ; (x -7.54276526206) (x -3.18185693794) = 0 ; ; ; ; : Nr. 1
c_1 = 2 * x_1 = 15.086 ; ; c_2 = 2 * x_2 = 6.364 ; ; : Nr. 1

4. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 7**2+15.09**2 - 2 * 7 * 15.09 * cos 40° } ; ; b = 10.71 ; ; : 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 = 7 ; ; b = 10.71 ; ; c = 15.09 ; ;

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

p = a+b+c = 7+10.71+15.09 = 32.8 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 32.8 }{ 2 } = 16.4 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.4 * (16.4-7)(16.4-10.71)(16.4-15.09) } ; ; T = sqrt{ 1151.84 } = 33.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 33.94 }{ 7 } = 9.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 33.94 }{ 10.71 } = 6.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 33.94 }{ 15.09 } = 4.5 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 10.71**2+15.09**2-7**2 }{ 2 * 10.71 * 15.09 } ) = 24° 49'58" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7**2+15.09**2-10.71**2 }{ 2 * 7 * 15.09 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 24° 49'58" - 40° = 115° 10'2" ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 33.94 }{ 16.4 } = 2.07 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7 }{ 2 * sin 24° 49'58" } = 8.33 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.71**2+2 * 15.09**2 - 7**2 } }{ 2 } = 12.607 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.09**2+2 * 7**2 - 10.71**2 } }{ 2 } = 10.469 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.71**2+2 * 7**2 - 15.09**2 } }{ 2 } = 5 ; ;
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