Triangle calculator

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

You have entered side a, b and height ha.

Triangle has two solutions: a=7.5; b=5.3; c=11.12877733506 and a=7.5; b=5.3; c=6.69772128723.

#1 Obtuse scalene triangle.

Sides: a = 7.5   b = 5.3   c = 11.12877733506

Area: T = 17.25
Perimeter: p = 23.92877733506
Semiperimeter: s = 11.96438866753

Angle ∠ A = α = 35.80110000919° = 35°48'4″ = 0.62548453271 rad
Angle ∠ B = β = 24.41773446738° = 24°25'2″ = 0.42661630592 rad
Angle ∠ C = γ = 119.7821655234° = 119°46'54″ = 2.09105842673 rad

Height: ha = 4.6
Height: hb = 6.50994339623
Height: hc = 3.11003507092

Median: ma = 7.86774118916
Median: mb = 9.11113209729
Median: mc = 3.34986064361

Inradius: r = 1.44218391337
Circumradius: R = 6.41105650824

Vertex coordinates: A[11.12877733506; 0] B[0; 0] C[6.82991892256; 3.11003507092]
Centroid: CG[5.98656541921; 1.03334502364]
Coordinates of the circumscribed circle: U[5.56438866753; -3.18441026586]
Coordinates of the inscribed circle: I[6.66438866753; 1.44218391337]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.1998999908° = 144°11'56″ = 0.62548453271 rad
∠ B' = β' = 155.5832655326° = 155°34'58″ = 0.42661630592 rad
∠ C' = γ' = 60.21883447657° = 60°13'6″ = 2.09105842673 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 = 7.5 ; ; b = 5.3 ; ; c = 11.13 ; ;

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

p = a+b+c = 7.5+5.3+11.13 = 23.93 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.93 }{ 2 } = 11.96 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.96 * (11.96-7.5)(11.96-5.3)(11.96-11.13) } ; ; T = sqrt{ 297.56 } = 17.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.25 }{ 7.5 } = 4.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.25 }{ 5.3 } = 6.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.25 }{ 11.13 } = 3.1 ; ;

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{ 7.5**2-5.3**2-11.13**2 }{ 2 * 5.3 * 11.13 } ) = 35° 48'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.3**2-7.5**2-11.13**2 }{ 2 * 7.5 * 11.13 } ) = 24° 25'2" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.13**2-7.5**2-5.3**2 }{ 2 * 5.3 * 7.5 } ) = 119° 46'54" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.25 }{ 11.96 } = 1.44 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.5 }{ 2 * sin 35° 48'4" } = 6.41 ; ;





#2 Acute scalene triangle.

Sides: a = 7.5   b = 5.3   c = 6.69772128723

Area: T = 17.25
Perimeter: p = 19.49772128723
Semiperimeter: s = 9.74986064361

Angle ∠ A = α = 76.44001541856° = 76°24'1″ = 1.33334342396 rad
Angle ∠ B = β = 43.38215010487° = 43°22'53″ = 0.75771500278 rad
Angle ∠ C = γ = 60.21883447657° = 60°13'6″ = 1.05110083863 rad

Height: ha = 4.6
Height: hb = 6.50994339623
Height: hc = 5.15113966568

Median: ma = 4.73437965871
Median: mb = 6.59876382235
Median: mc = 5.56438866753

Inradius: r = 1.76994836809
Circumradius: R = 3.85881769808

Vertex coordinates: A[6.69772128723; 0] B[0; 0] C[5.45109735355; 5.15113966568]
Centroid: CG[4.04993954693; 1.71771322189]
Coordinates of the circumscribed circle: U[3.34986064361; 1.91663414494]
Coordinates of the inscribed circle: I[4.44986064361; 1.76994836809]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 103.6599845814° = 103°35'59″ = 1.33334342396 rad
∠ B' = β' = 136.6188498951° = 136°37'7″ = 0.75771500278 rad
∠ C' = γ' = 119.7821655234° = 119°46'54″ = 1.05110083863 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 = 7.5 ; ; b = 5.3 ; ; c = 6.7 ; ;

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

p = a+b+c = 7.5+5.3+6.7 = 19.5 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 19.5 }{ 2 } = 9.75 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.75 * (9.75-7.5)(9.75-5.3)(9.75-6.7) } ; ; T = sqrt{ 297.56 } = 17.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.25 }{ 7.5 } = 4.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.25 }{ 5.3 } = 6.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.25 }{ 6.7 } = 5.15 ; ;

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{ 7.5**2-5.3**2-6.7**2 }{ 2 * 5.3 * 6.7 } ) = 76° 24'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.3**2-7.5**2-6.7**2 }{ 2 * 7.5 * 6.7 } ) = 43° 22'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.7**2-7.5**2-5.3**2 }{ 2 * 5.3 * 7.5 } ) = 60° 13'6" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.25 }{ 9.75 } = 1.77 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.5 }{ 2 * sin 76° 24'1" } = 3.86 ; ;




Look also our friend's collection of math examples and problems:

See more informations about triangles or more information about solving triangles.