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Squares Circumscribed by Circles

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Squares Circumscribed by Circles

Study Guide

Key Definition

When a square is circumscribed by a circle, the diagonal of the square is equal to the diameter of the circle. This can be expressed as $d = s\sqrt{2}$ where $d$ is the diagonal and $s$ is the side length of the square.

Important Notes

The diagonal of a square is $s\sqrt{2}$.
The diameter of the circle is the same as the diagonal of the square.
The radius of the circle is half of the diagonal.
Use the formula $A = \pi r^2$ to find the area of the circle.
The side length of a square can be found using $s = \frac{d}{\sqrt{2}}$.

Mathematical Notation

$\sqrt{2}$ represents the square root of 2

$\pi$ is approximately 3.14159

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

The diagonal of the square splits the square into two congruent right triangles, allowing the use of the Pythagorean theorem: $a^2 + b^2 = c^2$.

Remember

To find the side of a square when you know the diagonal, use $s = \frac{d}{\sqrt{2}}$.

Quick Reference

Diagonal Length:$d = s\sqrt{2}$

Circle Area:$A = \pi r^2$

Understanding Squares Circumscribed by Circles

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A square inside a circle touches the circle at the midpoints of its sides. The circle's diameter equals the square's diagonal, so if the diameter is d, then the side length is s = d/√2. The circle's area is A = πr², with r = d/2.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If a square is circumscribed by a circle with a diameter of $10$ cm, what is the side length of the square?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you have a square table that fits perfectly inside a circular room. The diameter of the room is $8$ meters. What is the side length of the table?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If the area of a circle circumscribing a square is $50\pi$ cm$^2$, what is the side length of the square?

Challenge Quiz

Single Choice Quiz

Advanced

A square is circumscribed by a circle with a radius of $7$ cm. What is the area of the square?

Recap

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Review key concepts and takeaways