Isosceles triangle calculator (S)

You have entered Area S and Angle γ.

Acute isosceles Triangle.

The lengths of the sides of the triangle:
a = 16.81879283051
b = 16.81879283051
c = 12.87218850581

Area: T = 100
Perimeter: p = 46.50877416683
Semiperimeter: s = 23.25438708341

Angle ∠ A = α = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ B = β = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Altitude to side a: h_a = 11.892207115
Altitude to side b: h_b = 11.892207115
Altitude to side c: h_c = 15.53877397403

Median: m_a = 12.3921666175
Median: m_b = 12.3921666175
Median: m_c = 15.53877397403

Inradius: r = 4.33003593128
Circumradius: R = 9.10217972112

Vertex coordinates: A[12.87218850581; 0] B[0; 0] C[6.43659425291; 15.53877397403]
Centroid: CG[6.43659425291; 5.17992465801]
Coordinates of the circumscribed circle: U[6.43659425291; 6.43659425291]
Coordinates of the inscribed circle: I[6.43659425291; 4.33003593128]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 112.5° = 112°30' = 1.96334954085 rad
∠ B' = β' = 112.5° = 112°30' = 1.96334954085 rad
∠ C' = γ' = 135° = 2.35661944902 rad

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How did we calculate this triangle?

The calculation of the triangle has two phases. The first phase calculates 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 triangle characteristics, 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: Angle γ and Area S

γ = 45 ° S = 100

2. From the Angle γ, we calculate Angle α:

α + β + γ = 180 ° α = β 2 α + γ = 180 ° α = 90 ° - γ / 2 = 90 ° - 45 ° / 2 = 67.5 °

3. From the Area S and Angle α, we calculate side c:

tan α = h : \frac{c}{2} h = \frac{c}{2} tan α T = \frac{c h}{2} T = \frac{c ^{2} tan α}{4} c = \frac{4 \cdot T}{tan α} = \frac{4 \cdot 1 0 0}{tan ( 6 7 . 5 ° )} = 12.872

4. From the Angle α, we calculate Angle β:

α = β β = 67.5 °

5. From the Angle α and side c, we calculate Altitude h_c:

tan α = h : c / 2 h = \frac{c}{2} \cdot tan α = \frac{1 2 . 8 7 2}{2} \cdot tan (67.5 °) = 15.538

6. From side c and altitude h, we calculate Side a - Pythagorean theorem:

a^{2} = h^{2} + (c / 2)^{2} a = h^{2} + (c / 2)^{2} = 15.53 8^{2} + (12.872 / 2)^{2} = 16.818

7. From the Side a, we calculate Side b:

b = a = 16.818

8. From the Side a and side c, we calculate Perimeter p:

p = 2 a + c = 2 \cdot 16.818 + 12.872 = 46.508

We know the lengths of all three sides of the triangle, so the triangle is uniquely specified. Next, we calculate another of its characteristics - the same procedure for calculating the triangle from the known three sides (SSS).

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

10. The semiperimeter of the triangle

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

11. Calculate the height of the isosceles triangle.

12. The triangle area

13. Calculation of the inner angles of the triangle - symmetry

14. Inradius

An incircle of a triangle is a tangent circle 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 a triangle's inradius and semiperimeter (half the perimeter) is its area.

15. 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. The circumcenter (center of the circumcircle) is the point where the perpendicular bisectors of a triangle intersect.

16. 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 of 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 a median's length from its side's lengths.

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