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### Course: High school geometry>Unit 8

Lesson 13: Constructing a line tangent to a circle

# Circles FAQ

## Why do we need to learn about circles?

There are many places where we use properties of circles in the real world. For example, architects often use arcs and sectors when designing buildings, and engineers use circles when designing gears and other mechanical parts. Additionally, understanding the properties of circles can help us solve problems in geometry, trigonometry, and calculus.

## What's the difference between a radius and a diameter?

The radius of a circle is the distance from the center of the circle to any point on the circle. The diameter is twice the radius, or the distance across the circle through the center.

## What are tangent and secant lines?

A tangent line is a line that touches a circle at just one point. A secant line is a line that cuts through a circle at two points.

Radians are a unit of measurement for angles.
One radian is the angle measure that we turn to travel one radius length around the circumference of a circle.
We often use radians in geometry because they make working certain formulas easier.

## What is arc length?

Arc length is the distance along a curved line, like on the edge of a circle.
On a circle, we need to know the central angle of the arc $\left(\theta \right)$ and the radius of the circle $\left(r\right)$ to find arc length.
If $\theta$ is in degrees:
$\text{arc length}=\frac{\theta }{360\mathrm{°}}\cdot 2\pi r$
If $\theta$ is in radians:
$\text{arc length}=\theta \cdot r$

## What is a sector?

A sector is a part of a circle, defined by two radii (the plural of radius) and the arc between them. Think of it as "pie slice" of the circle.

## How do we find the area of a sector?

We need to know the central angle of the sector $\left(\theta \right)$ and the radius of the circle $\left(r\right)$ to find the sector's area.
If $\theta$ is in degrees:
${\text{area}}_{\text{sector}}=\frac{\theta }{360\mathrm{°}}\cdot \pi {r}^{2}$
If $\theta$ is in radians:
${\text{area}}_{\text{sector}}=\frac{\theta }{2\pi }\cdot \pi {r}^{2}$
Note that we could simplify the formula that uses radians a bit:
${\text{area}}_{\text{sector}}=\frac{1}{2}\cdot \theta \cdot {r}^{2}$

## What's an inscribed angle?

An inscribed angle is an angle that has its vertex on the circumference of a circle.

## What is an inscribed shape?

An inscribed shape is a shape that is drawn inside a circle such that all of its vertices are touching the circumference of the circle.

## Want to join the conversation?

• I have some troubles on word problems. Any tips?
• I'm not good at math but here's some methods that I use when tackling word problems.

- Try to visualize the problem. Drawing out the problem statement tends to help you have a better grasp on it is trying to ask you.

- Write down every equation that you find. If you don't know the value of an object, simply let a variable to denote that unknown. This will let you have a (kinda) direction of what you will be trying to find.
For example if the problem states that "between two objects one is faster than another by 100 mph", you simply let one object's speed be A mph, another be B mph, and define A < B. Now you can write down A + 100 = B.
After you think you have written down all possible useful equations, go ahead and attempt connecting equations. But what formula to apply after this step requires you to be extremely familiar with all formulas (At least for the topic you are doing).

Note that this is my accumulated experience after doing a lot of problems. You should find a suitable way for you to solve problem comfortably, even if it means a lot of struggling.

Hope this helps.
• Does anyone else have trouble/tips with the area of a sector?