Equation of a Circle - GCSE Higher Maths

Summary

This video tutorial explains the equation of a circle, represented by x² + y² = r², with the center at the origin. It covers how to determine the radius from the equation, draw the corresponding graph, and convert various equations into the standard form. The presenter demonstrates examples with different radii, emphasizes understanding radius and diameter relationships, and teaches how to extract information from given equations. Additionally, it includes how to handle non-square numbers and mixed fractions. The tutorial concludes with questions about finding equations based on radius or diameter, enhancing the viewer's comprehension of circle geometry.

Takeaways

• 😀 The equation of a circle is represented as x² + y² = R², where R is the radius.
• 🎯 A circle centered at the origin will intersect the axes at ±R.
• 📝 To graph a circle, use a compass with one end at the center and the other at the radius length.
• 🔄 When given a circle's graph, determine its equation by identifying the radius and rewriting it in standard form.
• 📏 The radius can be found by rewriting the equation in the form x² + y² = r², where r is the radius.
• 🧮 The diameter of a circle is always double the radius.
• 🌐 The area of a circle is calculated using the formula A = πR².
• 🧩 If the equation includes a non-square number, express the radius as the square root of that number.
• 📊 For mixed numbers, convert them to improper fractions to simplify calculations.
• 💡 To find the equation of a circle from given dimensions, use the radius or diameter to form the equation x² + y² = r².

Q & A

What is the standard equation of a circle?

The standard equation of a circle is given by the formula x² + y² = r², where (0, 0) is the center and r is the radius.

How do you graph a circle using the equation x² + y² = 9?

To graph the equation x² + y² = 9, identify the radius as 3 (since √9 = 3) and draw a circle centered at the origin with a radius of 3 using a compass.

What is the relationship between the radius and diameter of a circle?

The diameter of a circle is always twice the radius. For instance, if the radius is 6, the diameter will be 12.

How can you determine the radius from the equation x² + y² = 36?

To find the radius from the equation x² + y² = 36, take the square root of 36, which gives a radius of 6.

What is the formula to calculate the area of a circle?

The area of a circle can be calculated using the formula Area = πr², where r is the radius.

How do you express the radius when the equation is not a perfect square, like x² + y² = 3?

When the equation is not a perfect square, express it as x² + y² = (√3)² to identify the radius as √3.

If a circle crosses the axes at (8, 0) and (0, 8), what is the equation of the circle?

The equation of the circle would be x² + y² = 8² = 64, indicating the radius is 8.

How do you find the equation of a circle if the diameter is given, say 24?

If the diameter is 24, the radius is half of that, which is 12. Thus, the equation is x² + y² = 12² = 144.

What does it mean to simplify the square root in the context of circle equations?

Simplifying the square root means expressing it in a more manageable form, such as √18 = √(9 × 2) = 3√2, which can be used to find the radius in the form a√b.

Why is it important to write circle equations in the form x² + y² = r²?

Writing circle equations in the form x² + y² = r² makes it easier to identify the center, radius, diameter, and calculate the area.