Circles

Study Guide

Key Definition

A circle is defined as the set of all points in a plane equidistant from a given point, known as the center. The distance from the center to any point on the circumference is referred to as the radius. A line segment connecting two points on the circumference of the circle and passing through its center is known as the diameter.

Important Notes

The diameter is twice the radius. This relationship can be expressed as $d = 2r$.
The circumference of a circle can be calculated using the formula $C = 2 \pi (r)$.

Mathematical Notation

$d$ represents the diameter of the circle.

$r$ represents the radius of the circle.

$\pi$ represents the mathematical constant Pi, approximately equal to 3.14.

Remember to use the correct symbols when solving problems related to circles.

Why It Works

The formulas for the diameter and circumference of a circle are derived from its definition and the mathematical constant Pi, which is the ratio of a circle's circumference to its diameter.

Remember

The radius is always half the diameter. The circumference is calculated by multiplying the diameter (or twice the radius) by Pi.

Quick Reference

Diameter:$d = 2r$

Circumference:$C = 2 \pi (r)$

Understanding Circles

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Video explanation of this concept

Beginner

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Beginner Explanation

A circle is a shape where all points are an equal distance from the center. The distance from the center to the edge is the radius, and the longest distance across the circle is the diameter, which is twice the radius.

Practice Problems

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Quick Quiz

Single Choice Quiz

Beginner

What is the circumference of a circle with a diameter of 10 units?

Real-World Problem

Question Exercise

Intermediate

Bicycle Wheel Scenario

Imagine a bicycle wheel has a diameter of 26 inches. How far will the bicycle travel after one full rotation of the wheel?

Advanced Challenge

Thinking Exercise

Advanced

Think About This

A circle has a radius of 4 meters. What is the longest straight-line distance you can walk within the circle?

Recap

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