An ellipse is a geometric shape defined as the set of all points where the sum of the distances from two fixed points (called the foci) is constant. The major axis is the longest diameter of the ellipse, and the minor axis is the shortest diameter.
Ellipse Formulas
Perimeter or Circumference of an Ellipse Formula:
The perimeter of an ellipse, denoted as P, represents the total length of its boundary. It can be approximated using three general formulas:
P ≈ π(a + b): Here, 'a' is the length of the semi-major axis, and 'b' is the length of the semi-minor axis. This formula provides a reasonable approximation for the perimeter.
Pythagorean Approximation: P ≈ π√[2(a² + b²)]: This formula offers another approximation by considering the sum of the squares of the semi-axes, providing a more accurate result than the first formula. This formula is an approximation based on the Pythagorean theorem, where the sum of the squares of 'a' and 'b' is taken, then squared and multiplied by π.
Ramanujan Approximation: P ≈ π[(3/2)(a + b) - √(ab)]: This formula provides an even more accurate approximation by taking into account the product of the semi-axes, which further refines the calculation.
Area of an Ellipse Formula:
The area of an ellipse, denoted as A, represents the total region enclosed by the ellipse. It can be calculated using the following general formula:
Area of ellipse = π * a * b: Here, 'a' represents the length of the semi-major axis, and 'b' represents the length of the semi-minor axis.
Eccentricity of an Ellipse Formula:
Eccentricity (e) of an ellipse quantifies its elongation and is defined as the ratio of the distance between one focus and the center of the ellipse to the distance between the center and a point on the ellipse. The eccentricity formula is given by:
e = √(1 - b²/a²), where 'a' is the length of the semi-major axis, and 'b' is the length of the semi-minor axis.
Latus Rectum of an Ellipse Formula:
The latus rectum of an ellipse represents a line drawn perpendicular to the transverse axis of the ellipse and passing through the foci of the ellipse. Its length, denoted as 'L,' can be calculated using the formula:
L = 2b²/a, where 'a' is the length of the semi-major axis, and 'b' is the length of the semi-minor axis.
Semi-Major Axis (a) Calculation:
- Formula: a = A / (π * b)
- Explanation: This formula allows you to find the length of the semi-major axis (a) of an ellipse when you know its area (A) and the length of the semi-minor axis (b). The semi-major axis is the longer of the two axes and represents the distance from the center of the ellipse to the farthest point along the major axis.
Semi-Minor Axis (b) Calculation:
- Formula: b = A / (π * a)
- Explanation: This formula helps you determine the length of the semi-minor axis (b) of an ellipse when you have its area (A) and the length of the semi-major axis (a). The semi-minor axis is the shorter of the two axes and represents the distance from the center of the ellipse to the farthest point along the minor axis.
Ellipse Calculations Formulas
Calculation examples using the formulas for the perimeter, area, eccentricity, and latus rectum of an ellipse, as well as the formulas to calculate the semi-major and semi-minor axes. Use of imperial units in these examples.
Example 1: Perimeter of an Ellipse
Suppose you have an ellipse with a semi-major axis (a) of 6 inches and a semi-minor axis (b) of 4 inches. Calculate the approximate perimeter using the three formulas.
Using the formula P ≈ π(a + b):
P ≈ π(6 inches + 4 inches) = π(10 inches) ≈ 31.42 inches
Using the formula P ≈ π√[2(a² + b²)]:
P ≈ π√[2(6 inches)² + (4 inches)²] = π√[72 square inches + 16 square inches] ≈ π√88 square inches ≈ 29.66 inches
Using the formula P ≈ π[(3/2)(a + b) - √(ab)]:
P ≈ π[(3/2)(6 inches + 4 inches) - √(6 inches * 4 inches)] = π[(3/2)(10 inches) - √24 square inches] ≈ π[(15 inches) - √24 square inches] ≈ π[15 inches - 4.899 inches] ≈ π[10.101 inches] ≈ 31.75 inches
Example 2: Area of an Ellipse
Let's say you have an ellipse with a semi-major axis (a) of 8 feet and a semi-minor axis (b) of 3 feet. Calculate the area using the formula:
Area of ellipse = πab
Area ≈ π * (8 feet) * (3 feet) ≈ π * 24 square feet ≈ 75.40 square feet
Example 3: Eccentricity of an Ellipse
For an ellipse with a semi-major axis (a) of 10 inches and a semi-minor axis (b) of 6 inches, calculate the eccentricity using the formula:
e = √(1 - b²/a²)
e = √(1 - (6 inches)² / (10 inches)²) = √(1 - 36/100) = √(0.64) ≈ 0.8
Example 4: Latus Rectum of an Ellipse
Given an ellipse with a semi-major axis (a) of 5 feet and a semi-minor axis (b) of 3 feet, calculate the latus rectum using the formula:
L = 2b²/a
L = 2 * (3 feet)² / (5 feet) = 18 square feet / 5 feet = 3.6 feet
Example 5: Semi-Major Axis Calculation
Suppose you know the area (A) of an ellipse is 50 square inches, and you have the length of the semi-minor axis (b) as 4 inches. Calculate the semi-major axis (a) using the formula:
a = A / (π * b)
a = 50 square inches / (π * 4 inches) ≈ 3.98 inches
Example 6: Semi-Minor Axis Calculation
If you have an ellipse with an area (A) of 120 square feet and you know the semi-major axis (a) is 10 feet, calculate the semi-minor axis (b) using the formula:
b = A / (π * a)
b = 120 square feet / (π * 10 feet) ≈ 3.82 feet