Circle Properties and Formulas

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6 Questions

What is the definition of a circle?

A closed shape in a two-dimensional plane where every point on the circle is equidistant from a fixed point called the center.

What is the longest distance across a circle, passing through its center?

Diameter

What is the formula for the area of a circle?

A = πr^2, where A is the area and r is the radius.

What are circles that share the same center called?

Concentric circles

What is the inscribed angle theorem?

The angle at the center of a circle is twice the angle at the circumference.

How are circles used in real-world applications?

Circles are used in geometry, trigonometry, and engineering to describe the shape of wheels, gears, and other circular objects, and to design circular structures such as bridges, tunnels, and pipes.

Study Notes

Definition

  • A circle is a closed shape in a two-dimensional plane that is every point on the circle is equidistant from a fixed point called the center.

Properties

  • Center: The fixed point equidistant from every point on the circle.
  • Radius: The distance from the center of the circle to any point on the circle.
  • Diameter: The longest distance across the circle, passing through its center.
  • Circumference: The distance around the circle.

Formulas

  • Circumference: C = 2πr, where C is the circumference and r is the radius.
  • Area: A = πr^2, where A is the area and r is the radius.

Types of Circles

  • Concentric circles: Circles that share the same center.
  • Congruent circles: Circles that have the same radius.

Circle Theorems

  • Inscribed angle theorem: The angle at the center of a circle is twice the angle at the circumference.
  • Central angle theorem: The angle at the center of a circle is equal to the sum of the two adjacent angles at the circumference.

Real-World Applications

  • Geometry: Circles are used to describe the shape of wheels, gears, and other circular objects.
  • Trigonometry: Circles are used to define trigonometric functions such as sine, cosine, and tangent.
  • Engineering: Circles are used in the design of circular structures such as bridges, tunnels, and pipes.

Circle Definition

  • A circle is a closed shape in a 2D plane where every point is equidistant from a fixed point called the center.

Circle Properties

  • The center is the fixed point equidistant from every point on the circle.
  • Radius is the distance from the center to any point on the circle.
  • Diameter is the longest distance across the circle, passing through its center.
  • Circumference is the distance around the circle.

Circle Formulas

  • Circumference (C) = 2πr, where r is the radius.
  • Area (A) = πr^2, where r is the radius.

Types of Circles

  • Concentric circles share the same center.
  • Congruent circles have the same radius.

Circle Theorems

  • Inscribed angle theorem: The angle at the center is twice the angle at the circumference.
  • Central angle theorem: The angle at the center equals the sum of the two adjacent angles at the circumference.

Real-World Applications

  • Geometry: Circles describe the shape of wheels, gears, and other circular objects.
  • Trigonometry: Circles define trigonometric functions like sine, cosine, and tangent.
  • Engineering: Circles are used in designing circular structures like bridges, tunnels, and pipes.

Learn about the definition and properties of a circle, including center, radius, diameter, and circumference, as well as formulas related to circles.

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