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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

5. 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{ 7**2-4.61**2-6.36**2 }{ 2 * 4.61 * 6.36 } ) = 77° 27'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4.61**2-7**2-6.36**2 }{ 2 * 7 * 6.36 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.36**2-7**2-4.61**2 }{ 2 * 4.61 * 7 } ) = 62° 32'50" ; ;

6. Inradius

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

7. Circumradius

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





#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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

5. 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{ 7**2-10.71**2-15.09**2 }{ 2 * 10.71 * 15.09 } ) = 24° 49'58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10.71**2-7**2-15.09**2 }{ 2 * 7 * 15.09 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15.09**2-7**2-10.71**2 }{ 2 * 10.71 * 7 } ) = 115° 10'2" ; ;

6. Inradius

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

7. Circumradius

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




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