Triangle calculator SSA

Please enter two sides and a non-included angle
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Triangle has two solutions with side c=95.06993151523 and with side c=59.08332067214

#1 Acute scalene triangle.

Sides: a = 89   b = 48   c = 95.06993151523

Area: T = 2115.292226214
Perimeter: p = 232.0699315152
Semiperimeter: s = 116.0354657576

Angle ∠ A = α = 67.98546304596° = 67°59'5″ = 1.18765556423 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 82.01553695404° = 82°55″ = 1.43114382357 rad

Height: ha = 47.53546575761
Height: hb = 88.13771775891
Height: hc = 44.5

Median: ma = 60.752226203
Median: mb = 88.90221222568
Median: mc = 53.41330726426

Inradius: r = 18.23298315549
Circumradius: R = 48

Vertex coordinates: A[95.06993151523; 0] B[0; 0] C[77.07662609368; 44.5]
Centroid: CG[57.38218586964; 14.83333333333]
Coordinates of the circumscribed circle: U[47.53546575761; 6.66875579578]
Coordinates of the inscribed circle: I[68.03546575761; 18.23298315549]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 112.015536954° = 112°55″ = 1.18765556423 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 97.98546304596° = 97°59'5″ = 1.43114382357 rad


How did we calculate this triangle?

The calculation of the triangle progress in two phases. The first phase is such that we 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 is the calculation of 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. Use the Law of Cosines

a = 89; b = 48; beta = 30°; ; ; b**2 = a**2 + c**2 - 2ac cos beta; 48**2 = 89**2 + c**2 -2 * 89 * c * cos (30° ); ; ; c**2 -154.153c +5617 =0; ; ; p=1; q=-154.153; r=5617; D = q**2 - 4pr = 154.153**2 - 4 * 1 * 5617 = 1295
D>0; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 154.15 ± sqrt{ 1295 } }{ 2 }; c_{1,2} = 77.07626094 ± 17.9930542154; c_{1} = 95.0693151523; c_{2} = 59.0832067214; ; ; text{ Factored form of the equation: }; (c -95.0693151523) (c -59.0832067214) = 0
; c>0
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 = 89; b = 48; c = 95.07

2. The triangle perimeter is the sum of the lengths of its three sides

p = a+b+c = 89+48+95.07 = 232.07

3. 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.

s = fraction{ p }{ 2 } = fraction{ 232.07 }{ 2 } = 116.03

4. The triangle area using Heron's formula

Heron's formula gives the area of a triangle when the length of all three sides are 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.

T = sqrt{ s(s-a)(s-b)(s-c) }; T = sqrt{ 116.03 * (116.03-89)(116.03-48)(116.03-95.07) }; T = sqrt{ 4474461.35 } = 2115.29

5. 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 length of the base and the 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.

T = fraction{ a h_a }{ 2 }; ; ; h_a = fraction{ 2 T }{ a } = fraction{ 2 * 2115.29 }{ 89 } = 47.53; h_b = fraction{ 2 T }{ b } = fraction{ 2 * 2115.29 }{ 48 } = 88.14; h_c = fraction{ 2 T }{ c } = fraction{ 2 * 2115.29 }{ 95.07 } = 44.5

6. Calculation of the inner angles of the triangle using a Law of Cosines

The Law of Cosines is useful for finding the angles of a triangle 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 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 alpha; ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 48**2+95.07**2-89**2 }{ 2 * 48 * 95.07 } ) = 67° 59'5"; ; ; b**2 = a**2+c**2 - 2ac cos beta; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 89**2+95.07**2-48**2 }{ 2 * 89 * 95.07 } ) = 30°
 gamma = 180° - alpha - beta = 180° - 67° 59'5" - 30° = 82° 55"

7. Inradius

An incircle of a triangle is a circle which is tangent to each side. An incircle center is called 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.

T = rs; r = fraction{ T }{ s } = fraction{ 2115.29 }{ 116.03 } = 18.23

8. Circumradius

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 = fraction{ a b c }{ 4 r s } = fraction{ 89 * 48 * 95.07 }{ 4 * 18.23 * 116.035 } = 48

9. Calculation of medians

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. 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.

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 95.07**2 - 89**2 } }{ 2 } = 60.752; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 95.07**2+2 * 89**2 - 48**2 } }{ 2 } = 88.902; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 89**2 - 95.07**2 } }{ 2 } = 53.413



#2 Obtuse scalene triangle.

Sides: a = 89   b = 48   c = 59.08332067214

Area: T = 1314.601134955
Perimeter: p = 196.0833206721
Semiperimeter: s = 98.04216033607

Angle ∠ A = α = 112.015536954° = 112°55″ = 1.95550370113 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 37.98546304596° = 37°59'5″ = 0.66329568667 rad

Height: ha = 29.54216033607
Height: hb = 54.77550562313
Height: hc = 44.5

Median: ma = 30.28546934645
Median: mb = 71.62334085913
Median: mc = 65.1143698028

Inradius: r = 13.40986072085
Circumradius: R = 48

Vertex coordinates: A[59.08332067214; 0] B[0; 0] C[77.07662609368; 44.5]
Centroid: CG[45.38664892194; 14.83333333333]
Coordinates of the circumscribed circle: U[29.54216033607; 37.83224420422]
Coordinates of the inscribed circle: I[50.04216033607; 13.40986072085]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 67.98546304596° = 67°59'5″ = 1.95550370113 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 142.015536954° = 142°55″ = 0.66329568667 rad

Calculate another triangle

How did we calculate this triangle?

The calculation of the triangle progress in two phases. The first phase is such that we 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 is the calculation of 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. Use the Law of Cosines

a = 89; b = 48; beta = 30°; ; ; b**2 = a**2 + c**2 - 2ac cos beta; 48**2 = 89**2 + c**2 -2 * 89 * c * cos (30° ); ; ; c**2 -154.153c +5617 =0; ; ; p=1; q=-154.153; r=5617; D = q**2 - 4pr = 154.153**2 - 4 * 1 * 5617 = 1295 : Nr. 1
D>0; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 154.15 ± sqrt{ 1295 } }{ 2 }; c_{1,2} = 77.07626094 ± 17.9930542154; c_{1} = 95.0693151523; c_{2} = 59.0832067214; ; ; text{ Factored form of the equation: }; (c -95.0693151523) (c -59.0832067214) = 0 : Nr. 1
; c>0 : 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 = 89; b = 48; c = 59.08

2. The triangle perimeter is the sum of the lengths of its three sides

p = a+b+c = 89+48+59.08 = 196.08

3. 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.

s = fraction{ p }{ 2 } = fraction{ 196.08 }{ 2 } = 98.04

4. The triangle area using Heron's formula

Heron's formula gives the area of a triangle when the length of all three sides are 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.

T = sqrt{ s(s-a)(s-b)(s-c) }; T = sqrt{ 98.04 * (98.04-89)(98.04-48)(98.04-59.08) }; T = sqrt{ 1728176.71 } = 1314.6

5. 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 length of the base and the 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.

T = fraction{ a h_a }{ 2 }; ; ; h_a = fraction{ 2 T }{ a } = fraction{ 2 * 1314.6 }{ 89 } = 29.54; h_b = fraction{ 2 T }{ b } = fraction{ 2 * 1314.6 }{ 48 } = 54.78; h_c = fraction{ 2 T }{ c } = fraction{ 2 * 1314.6 }{ 59.08 } = 44.5

6. Calculation of the inner angles of the triangle using a Law of Cosines

The Law of Cosines is useful for finding the angles of a triangle 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 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 alpha; ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 48**2+59.08**2-89**2 }{ 2 * 48 * 59.08 } ) = 112° 55"; ; ; b**2 = a**2+c**2 - 2ac cos beta; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 89**2+59.08**2-48**2 }{ 2 * 89 * 59.08 } ) = 30°
 gamma = 180° - alpha - beta = 180° - 112° 55" - 30° = 37° 59'5"

7. Inradius

An incircle of a triangle is a circle which is tangent to each side. An incircle center is called 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.

T = rs; r = fraction{ T }{ s } = fraction{ 1314.6 }{ 98.04 } = 13.41

8. Circumradius

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 = fraction{ a b c }{ 4 r s } = fraction{ 89 * 48 * 59.08 }{ 4 * 13.409 * 98.042 } = 48

9. Calculation of medians

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. 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.

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 59.08**2 - 89**2 } }{ 2 } = 30.285; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 59.08**2+2 * 89**2 - 48**2 } }{ 2 } = 71.623; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 89**2 - 59.08**2 } }{ 2 } = 65.114
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