Main content

### Course: 7th grade > Unit 9

Lesson 1: Area and circumference of circles- Geometry FAQ
- Radius, diameter, circumference & π
- Labeling parts of a circle
- Radius, diameter, & circumference
- Radius and diameter
- Radius & diameter from circumference
- Relating circumference and area
- Circumference of a circle
- Area of a circle
- Area of a circle
- Partial circle area and arc length
- Circumference of parts of circles
- Area of parts of circles
- Circumference review
- Area of circles review

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Partial circle area and arc length

Sal finds the area of a semicircle and the arc length of a partial circle.

## Want to join the conversation?

- At1:07to2:59, couldn't he find the area and then multiply it by 3/4 and you get an answer?(32 votes)
- Sure, if what you needed was the area. But the exercise is asking for the arc length which is part of the circumference and not the area. Just remember that arc length, and circumference for that matter, is a measurement of length as the name implies while area is not.(51 votes)

- does anyone have a word problem example for this? I believe it could help me comprehend this a little more.(11 votes)
- 1st problem:

You need to bake a circular cake for a birthday party, the radius of the cake is 2dm (=20cm).

Looking from above, you want half the circle to be covered in ice sugar, the other half will be covered in chocolate powder.

Now you want to go an buy the ice sugar. At the shop, the ice sugar is sold in little bags for "1 dm2 cake coverage" (weird shop, I know, but it's the only one open :-) )

How many bags do you need to buy?

To answer, the easiest is to calculate the surface area of the whole cake, then divide it by 2 (because the rest will be covered in chocolate)

Surface area of the whole cake: Pi*(radius)squared = Pi*(2)squared = 4*Pi,

Half of the cake's surface area = (4*Pi)/2 = 2*Pi = approx. 6.28dm2,

So you'll need at least 7 bags to cover the part you want to cover.

For the second problem:

Now you decide you want to put the cake on a wooden circular stand. The stand's radius is 4 dm. You want to decorate all around the perimeter of the stand. You're thinking of using golden paper for 3/4 of the circumference, then you'll use red paper for the rest (1/4 of the circumference)

Now you want to purchase some golden paper. What length do you need to buy?

To calculate the length (= the arc length) you need to calculate the whole circumference first and then multiply it by 3/4 to find the length you really want:

Whole circumference = 2*Pi*radius = 8*Pi = approx. 25.13dm,

The length that you need: (8*Pi) * (3/4 ) = 6*Pi = approx. 18.85dm.

Hope that helps!(31 votes)

- Is it the area or the circumference we're finding?

And rather than 2 pi r i'd use pi d. Is that okay?(16 votes)- the first problem was area,but the second problem was circumference since it was arc length.Pi times D is fine (that's what I would have done)(14 votes)

- still couldn't find the arc after 7 tries(16 votes)
- "Try Try but don't cry"

Always remember this quote and go forward!

:) :>(9 votes)

- i still dont know how to do this(14 votes)
- Hey there!

(Don't get scared! I know the answer is quite long, but very very simple)

There's a generalized formula for calculating the arc---very simple when you realize the steps, I have derived it in the form of proportion:

let x be the arc we are searching for

C be the circumference

C=2*pi*radius

A is the central angle(angle formed by the 2

radii)

x:A::C:360 === 'arc' = C when central angle is 360 deg (complete circle), what would be arc if angle = A

**x=(C*A)/360**

That's it!

if you're getting confused by the terms, let me help you out here,*Circumference*= perimeter of the circle*arc*= part of the circumference lying between 2

radii(plural form of radius).*central angle*= angle formed by two radii...where? at the centre

a question may arise here asking which central angle corresponds to which arc

the larger the angle=larger the arc, you can then correlate.

in one of the questions he mentioned above, the area(Pi*r^2) is given,

to find the circumference we need to know the value of r(the only variable) then we can calculate the arc.

Hope you understood and I hope I cleared your doubt but @25jagoins, it would be great if you state**what**you didn't understand and be a bit less vague while asking questions, it would help people help you.

Keep up your consistency and don't give up!

Onward!(8 votes)

- "So pause this video and see if you can figure it out." Sal, I'm here to figure out HOW to solve these, not just guess and hope I'm right(15 votes)
- How do you find the quarter of a circles area(8 votes)
- To find the area of a quarter circle, find the area of the whole circle by using the formula A = pi * r^2 and then divide by 4

One-quarter of a circle is called a quadrant. According to Reference.com, it is also a 90 degree arc. "Quad" is a prefix that means "fourth.(2 votes)

- queen elizabeth died yesterday(11 votes)
- At2:12, why do you multiply 3/4?(2 votes)
- Because the formula 2 π r is the formula for the circumference of the WHOLE circle. But they only want the arc length of 3/4 of the circle. So the answer of 8π was the answer for the whole circle so you have to multiply it by 3/4 which gives you 6π, The answer to the arc length that they asked for. :)(13 votes)

- Was this a Pokémon reference on0:48?(9 votes)
- wow how smart, you get an upvote(0 votes)

## Video transcript

- [Instructor] Find the
area of the semicircle. So pause this video and see
if you can figure it out. So let's see. We know that the area of a circle is equal to pi times our radius squared. So, if we think about the entire circle, what is the area going to be? Well, they tell us what our radius is. Our radius is equal to two, so the area, if we're talking
about the whole circle, it would be equal to pi times two squared. Pi times two squared. Two squared is of course two times two, which is equal to four, so our area is going to
be equal to four times pi. Now, I wouldn't put four pi here, because that would be the entire circle. They want the area of just the semicircle, of just this region right over here. Well, the semicircle
is half of the circle, so if I want the area of the semicircle, this is gonna be half this. So instead of four pi, it is going to be two pi square units. That's the area of the semicircle. Let's do another example. So here, instead of area, we're asked to find the arc
length of the partial circle, and that's we have here
in this bluish color right over here, find this arc length. And you can see this
is going three fourths of the way around the circle, so this arc length is going to be three fourths of the circumference. So what is the circumference? Well, we know the circumference is equal to two pi times the radius. They tell us what the radius is. It's equal to four, so our circumference is equal to two pi times four. Let's see, we can just change
the order in which we multiply so it's two times four times pi. This is going to be equal to eight pi. This is going to be equal to eight pi. Now, that is the circumference
of the entire circle. If we care about this arc length, it's going to be three fourths times the circumference of the entire circle. So three over four times eight pi. What is that going to be? Well, what's three fourths times eight? Well, three times eight is
24 divided by four is six. So this is going to be equal to six pi. Another way to think about it, one fourth of eight is two, so three fourths is going to be six. Or another way to think about it is, one fourth of eight pi is two pi, and so three of those is
going to be equal to six pi. So the arc length of the
partial circle is six pi, and once again we knew that because it was three
fourths of the way around. The way that I knew it was three fourths is that this is a 90 degree angle. This is 90 degrees, which is one fourth of the way around a circle, so the arc length that we care about is the three fourths of our circumference.