# Triangle calculator - the result

You have entered side a, b, and angle α.

Triangle has two solutions: a=60; b=102; c=56.72876299275 and a=60; b=102; c=119.9421552445.

### #1 Obtuse scalene triangle.

Sides: a = 60   b = 102   c = 56.72876299275

Area: T = 1446.555456315
Perimeter: p = 218.7287629928
Semiperimeter: s = 109.3643814964

Angle ∠ A = α = 30° = 0.52435987756 rad
Angle ∠ B = β = 121.7888330617° = 121°47'18″ = 2.12656073598 rad
Angle ∠ C = γ = 28.21216693829° = 28°12'42″ = 0.49223865182 rad

Height: ha = 48.21884854383
Height: hb = 28.36438149637
Height: hc = 51

Median: ma = 76.88331060675
Median: mb = 28.42655518608
Median: mc = 78.72441640204

Vertex coordinates: A[56.72876299275; 0] B[0; 0] C[-31.60769612586; 51]
Centroid: CG[8.3743556223; 17]
Coordinates of the circumscribed circle: U[28.36438149637; 52.87224313863]
Coordinates of the inscribed circle: I[7.36438149637; 13.22769943549]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 150° = 0.52435987756 rad
∠ B' = β' = 58.21216693829° = 58°12'42″ = 2.12656073598 rad
∠ C' = γ' = 151.7888330617° = 151°47'18″ = 0.49223865182 rad

# How did we calculate this triangle?

The calculation of the triangle progress in two phases. The first phase is to try to calculate all three sides of the triangle from the input parameters. The first phase is different for the different triangles query entered. The second phase calculates other characteristics of the triangle, such as angles, area, perimeter, heights, the center of gravity, circle radii, etc. Some input data also results in two to three correct triangle solutions (e.g., if the specified triangle area and two sides - typically resulting in both acute and obtuse) triangle).

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

$a = 60 \ \\ b = 102 \ \\ α = 30^\circ$

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

Now we know the lengths of all three sides of the triangle, and the triangle is uniquely determined. Next, we calculate another of its characteristics - the same procedure as calculation of the triangle from the known three sides SSS.

$a = 60 \ \\ b = 102 \ \\ c = 56.73$

### 4. Semiperimeter of the triangle

The semiperimeter of the triangle is half its perimeter. The semiperimeter frequently appears in formulas for triangles that it is given a separate name. By the triangle inequality, the longest side length of a triangle is less than the semiperimeter.

### 5. The triangle area using Heron's formula

Heron's formula gives the area of a triangle when the length of all three sides is known. There is no need to calculate angles or other distances in the triangle first. Heron's formula works equally well in all cases and types of triangles.

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

There are many ways to find the height of the triangle. The easiest way is from the area and base length. The area of a triangle is half of the product of the base's length and height. Every side of the triangle can be a base; there are three bases and three heights (altitudes). Triangle height is the perpendicular line segment from a vertex to a line containing the base.

### 7. Calculation of the inner angles of the triangle using a Law of Cosines

The Law of Cosines is useful for finding a triangle's angles when we know all three sides. The cosine rule, also known as the law of cosines, relates all three sides of a triangle with an angle of a triangle. The Law of Cosines is the extrapolation of the Pythagorean theorem for any triangle. Pythagorean theorem works only in a right triangle. Pythagorean theorem is a special case of the Law of Cosines and can be derived from it because the cosine of 90° is 0. It is best to find the angle opposite the longest side first. With the Law of Cosines, there is also no problem with obtuse angles as with the Law of Sines because the cosine function is negative for obtuse angles, zero for right, and positive for acute angles. We also use inverse cosine called arccosine to determine the angle from cosine value.

$a^2 = b^2+c^2 - 2bc \cos α \ \\ \ \\ α = \arccos(\dfrac{ b^2+c^2-a^2 }{ 2bc } ) = \arccos(\dfrac{ 102^2+56.73^2-60^2 }{ 2 \cdot \ 102 \cdot \ 56.73 } ) = 30^\circ \ \\ \ \\ b^2 = a^2+c^2 - 2ac \cos β \ \\ β = \arccos(\dfrac{ a^2+c^2-b^2 }{ 2ac } ) = \arccos(\dfrac{ 60^2+56.73^2-102^2 }{ 2 \cdot \ 60 \cdot \ 56.73 } ) = 121^\circ 47'18" \ \\ γ = 180^\circ - α - β = 180^\circ - 30^\circ - 121^\circ 47'18" = 28^\circ 12'42"$

An incircle of a triangle is a circle which is tangent to each side. An incircle center is called an incenter and has a radius named inradius. All triangles have an incenter, and it always lies inside the triangle. The incenter is the intersection of the three-angle bisectors. The product of the inradius and semiperimeter (half the perimeter) of a triangle is its area.

The circumcircle of a triangle is a circle that passes through all of the triangle's vertices, and the circumradius of a triangle is the radius of the triangle's circumcircle. Circumcenter (center of circumcircle) is the point where the perpendicular bisectors of a triangle intersect.

$R = \dfrac{ a b c }{ 4 \ r s } = \dfrac{ 60 \cdot \ 102 \cdot \ 56.73 }{ 4 \cdot \ 13.227 \cdot \ 109.364 } = 60$

### 10. Calculation of medians

A median of a triangle is a line segment joining a vertex to the opposite side's midpoint. Every triangle has three medians, and they all intersect each other at the triangle's centroid. The centroid divides each median into parts in the ratio 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex. We use Apollonius's theorem to calculate the length of a median from the lengths of its side.

### #2 Obtuse scalene triangle.

Sides: a = 60   b = 102   c = 119.9421552445

Area: T = 3058.510958734
Perimeter: p = 281.9421552445
Semiperimeter: s = 140.9710776222

Angle ∠ A = α = 30° = 0.52435987756 rad
Angle ∠ B = β = 58.21216693829° = 58°12'42″ = 1.01659852938 rad
Angle ∠ C = γ = 91.78883306171° = 91°47'18″ = 1.60220085842 rad

Height: ha = 101.9550319578
Height: hb = 59.97107762223
Height: hc = 51

Median: ma = 107.2154681837
Median: mb = 79.95499093271
Median: mc = 58.35767134038

Vertex coordinates: A[119.9421552445; 0] B[0; 0] C[31.60769612586; 51]
Centroid: CG[50.51661712344; 17]
Coordinates of the circumscribed circle: U[59.97107762223; -1.87224313863]
Coordinates of the inscribed circle: I[38.97107762223; 21.69660540993]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 150° = 0.52435987756 rad
∠ B' = β' = 121.7888330617° = 121°47'18″ = 1.01659852938 rad
∠ C' = γ' = 88.21216693829° = 88°12'42″ = 1.60220085842 rad

# How did we calculate this triangle?

The calculation of the triangle progress in two phases. The first phase is to try to calculate all three sides of the triangle from the input parameters. The first phase is different for the different triangles query entered. The second phase calculates other characteristics of the triangle, such as angles, area, perimeter, heights, the center of gravity, circle radii, etc. Some input data also results in two to three correct triangle solutions (e.g., if the specified triangle area and two sides - typically resulting in both acute and obtuse) triangle).

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

$a = 60 \ \\ b = 102 \ \\ α = 30^\circ$

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

Now we know the lengths of all three sides of the triangle, and the triangle is uniquely determined. Next, we calculate another of its characteristics - the same procedure as calculation of the triangle from the known three sides SSS.

$a = 60 \ \\ b = 102 \ \\ c = 119.94$

### 4. Semiperimeter of the triangle

The semiperimeter of the triangle is half its perimeter. The semiperimeter frequently appears in formulas for triangles that it is given a separate name. By the triangle inequality, the longest side length of a triangle is less than the semiperimeter.

### 5. The triangle area using Heron's formula

Heron's formula gives the area of a triangle when the length of all three sides is known. There is no need to calculate angles or other distances in the triangle first. Heron's formula works equally well in all cases and types of triangles.

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

There are many ways to find the height of the triangle. The easiest way is from the area and base length. The area of a triangle is half of the product of the base's length and height. Every side of the triangle can be a base; there are three bases and three heights (altitudes). Triangle height is the perpendicular line segment from a vertex to a line containing the base.

### 7. Calculation of the inner angles of the triangle using a Law of Cosines

The Law of Cosines is useful for finding a triangle's angles when we know all three sides. The cosine rule, also known as the law of cosines, relates all three sides of a triangle with an angle of a triangle. The Law of Cosines is the extrapolation of the Pythagorean theorem for any triangle. Pythagorean theorem works only in a right triangle. Pythagorean theorem is a special case of the Law of Cosines and can be derived from it because the cosine of 90° is 0. It is best to find the angle opposite the longest side first. With the Law of Cosines, there is also no problem with obtuse angles as with the Law of Sines because the cosine function is negative for obtuse angles, zero for right, and positive for acute angles. We also use inverse cosine called arccosine to determine the angle from cosine value.

$a^2 = b^2+c^2 - 2bc \cos α \ \\ \ \\ α = \arccos(\dfrac{ b^2+c^2-a^2 }{ 2bc } ) = \arccos(\dfrac{ 102^2+119.94^2-60^2 }{ 2 \cdot \ 102 \cdot \ 119.94 } ) = 30^\circ \ \\ \ \\ b^2 = a^2+c^2 - 2ac \cos β \ \\ β = \arccos(\dfrac{ a^2+c^2-b^2 }{ 2ac } ) = \arccos(\dfrac{ 60^2+119.94^2-102^2 }{ 2 \cdot \ 60 \cdot \ 119.94 } ) = 58^\circ 12'42" \ \\ γ = 180^\circ - α - β = 180^\circ - 30^\circ - 58^\circ 12'42" = 91^\circ 47'18"$

An incircle of a triangle is a circle which is tangent to each side. An incircle center is called an incenter and has a radius named inradius. All triangles have an incenter, and it always lies inside the triangle. The incenter is the intersection of the three-angle bisectors. The product of the inradius and semiperimeter (half the perimeter) of a triangle is its area.

The circumcircle of a triangle is a circle that passes through all of the triangle's vertices, and the circumradius of a triangle is the radius of the triangle's circumcircle. Circumcenter (center of circumcircle) is the point where the perpendicular bisectors of a triangle intersect.

$R = \dfrac{ a b c }{ 4 \ r s } = \dfrac{ 60 \cdot \ 102 \cdot \ 119.94 }{ 4 \cdot \ 21.696 \cdot \ 140.971 } = 60$

### 10. Calculation of medians

A median of a triangle is a line segment joining a vertex to the opposite side's midpoint. Every triangle has three medians, and they all intersect each other at the triangle's centroid. The centroid divides each median into parts in the ratio 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex. We use Apollonius's theorem to calculate the length of a median from the lengths of its side.