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### Course: Geometry (all content)>Unit 7

Lesson 8: Area and circumference of circles

# Area of a circle intuition

Using triangles to create an informal argument for the area of a circle formula.

## Want to join the conversation?

• I know this is off topic, but what is a seven sided polygon called?
• 7 sides is a heptagon.
• Hi! Why did he divide the 2(pi)r*r expression at the end by two? Thanks you :^)
• Hi. This expression was divided by two at the end because the original expression was n(b x a/2), which can also be written as just n x b x a x 1/2 using properties of multiplication. n x b, the number of sides of the polygon multiplied by the length of those sides (the perimeter of the polygon) gets closer to the circle's circumference as n, the number of sides, approaches infinity. You can imagine adding more sides to the polygon and watch the perimeter getting so close to the circumference of the circle, but never quite get there since you would need an infinite number of sides.
The circumference is equal to 2 x (pi) x r, so you can replace n x b with 2 x (pi) x r in our new equation where n is essentially infinity, and the perimeter of the polygon becomes the circumference of the circle. a gets closer to the radius of the circle as n increases, so you can replace a with r, and the 1 / 2 stays the same. The whole expression becomes 2 x (pi) x r x r x 1 / 2, the 2 and 1 / 2 cancel out, and the expression simplifies to just *(pi) x r x r*, or *(pi)r^2*, the formula for the area of a circle.

I hope this helps! :) I think your confusion was just from not quite understanding that the expression was just n(b x a/2) with the variables replaced with what they approach as n approaches infinity, and that expression contained a 1 / 2 from the area for a triangle formula, which Sal just wrote by dividing the rest of the expression by two.
• Even if n approaches infinity the area of the circle won't be extremely precise. Wouldn't it be off by a little
• In the sense that we can only evaluate the area for finite 𝑛, the answer is yes, we will always be a little off even for very large 𝑛. However, the concept Sal is getting at here is the notion of a limit. That is, what does some function approach as the input approaches some value? In this case, even though the area function only approximates the area of a circle for all finite 𝑛, we can take the limit as 𝑛 → ∞ which will give us the area of the shape that this process is approaching as we increase 𝑛 (namely a circle). The concept of the limit is easily misunderstood when it is first introduced and there are rigorous definitions that make it even harder to understand, but hopefully you see the intuition behind Sal's logic. We are not looking at subcases for finite 𝑛, rather we are seeing what the area approaches as 𝑛 gets arbitrarily large (𝑛 → ∞).
Also, I highly recommend to anyone who asks about calculus that they focus on algebra, trigonometry, and geometry first before studying calculus. So don't worry about limits now if you haven't made yourself familiar with everything that precedes it.
Comment if you have questions!
• so theoretically a circle is just an infinite sided polygon?
• Infinity is not a number, it is a concept, so having infinite sides has no real meaning. You have to go to calculus and limits to sort of get to where you want to get.
• But saying that the area of a circle is approaching πr^2 is different than saying: the area IS πr^2...right?
• This is better understood when you take precalculus because Sal is talking about limits. Sal states that as n approaches infinity, the area approaches πr^2 which is a true statement, as you add more and more sides to the polygon, the area gets closer and closer to the formula. If there were such a thing as an infinite number of sides (a circle does not have sides) which would mean each point on the circle were considered a side, then it would reach the area formula. The theory might be beyond what you have seen in math before.
• At about minutes into the video, Sal said, n*b is approaching the circumference, but how is this possible?
• hi, I'm here to help. when sal said n*b is approaching the circumference he also mentioned when n approaches infinity and multiplied by b it will approach the circumference since the space will become 0.I hope that I have helped you.If not ask Sal.
• When all those properties approach the properties of a circle, does it ever become it?
• No. For a circle to be perfect, we would need to measure an infinite number of points around the circle's circumference to know for sure. A perfect circle is impossible to achieve in real life. From subatomic particles to carefully built structures, nothing in the physical world can be a perfect circle.
(1 vote)
• i dont get this