What is the EXACT AREA of a ring that has an inside radius of 6 cm and an outside radius of 12 cm?

[@gavindeane3670]

From the diagram it's not completely clear whether 6cm is the radius or diameter of the inner circle. Video description says it's the radius. So the yellow shaded area is π×12² - π×6² (144 - 36)π 108π cm² That's a little over 339 cm², but the video description asks for the exact answer, which is 108π cm².

[@panlomito]

A of a circle is A = pi . r² A out = pi . 12² = 144 . pi and A in = pi . 6² = 36 . pi A ring = A out - A in = pi . (144 - 36) = 108 . pi cm²

[@luillierstephane1463]

108pi•cm², or approximately 339.3 cm².

[@pas6295]

If u need the area of the green shaded one then it is area of the bigger circle minus the area of Circular Ring. Hence the answer.

[@pas6295]

Inner circle DIAMETER is 6 CMS. So RADIUS 3 CMS. SIMILAR oUTER which has 12 cms has Radius as 6 cms. Formula for the Circular RING with The Same Center is PAI(R+r)(R-r). = 9•3=27•3.414 SQ.CMS.

[@gavindeane3670]

The 6cm annotation is a little unclear as to whether it refers to the radius of diameter of the inner circle, but the 12cm annotation is very clearly the radius, not the diameter. The video description makes it clear that both are radius.

[@stevendebettencourt7651]

We’re starting to enter some trickier worlds of mathematics, but don’t be scared! We shall journey through these new lands together. Now, we may not know the area of a ring, but I will assume you know the formula for the area of a circle. If not, it is: A = pi*r^2, where A is the area of the circle, r is the radius of the circle, and pi is the ratio of a circle’s perimeter to its diameter (about 3.14) Notice that a ring consists of a large circle with a small circle removed from it. Then the area of the ring, A(R), is given by: A(R) = A(C) - A(c), with A(C) and A(c) being the areas of the large and small circles, respectively. Using R and r for the radii of these two circles, we get this formula: A(C) - A(c) = pi*R^2 - pi*r^2 = pi(R^2 - r^2) Notice this becomes pi(R + r)(R - r) if you don’t want to deal with squaring, but we’re gonna go ahead with squaring. In this case, R = 12 cm and r = 6 cm. Therefore: A(R) = pi((12 cm)^2 - (6 cm)^2) = pi(144 sq cm - 36 sq cm) = pi(108 sq cm) = 108*pi square centimeters In general, unless the problem says otherwise, it’s usually okay to keep pi in your answer due to its nature. If not, make sure you use the correct number of significant digits. That wasn’t too bad, but let’s imagine taking this to the next level: Put the top half of this ring on an x-y coordinate grid and then imagine rotating the area around a certain axis until it meets itself again. You’ve now created a 3D solid shape with some volume. Generally, you would use calculus to calculate the volume of this shape, but we’ll cross that bridge another day.

[@mauriziograndi1750]

I resolved the two different areas and subtracted them = 339.12 square cm.

[@richardkayebloede1070]

I agree with you

[@CC-iq2pe]

144pi cm2 - 36pi cm2

[@russelllomando8460]

got 108 Pi cm^2 144 - 36 = 108 thanks for the fun.

[@dankrukov1890]

~324

[@robertsmith4466]

I calculated the area to 339 3/7 which is also exact I think?

[@gavindeane3670]

No, it's not exact. We are dealing with areas of circles so the answer will involve pi. Pi is an irrational number so there is no fractional or decimal representation of the exact answer.

[@robertsmith4466]

@@gavindeane3670 Pi is 22/7 that is the exact value 339 3/7 is also an exact answer to this calculation. It may be the convention to use Pi but it is no more exact in my opinion. What proof can you give me to change my mind?

[@richardkayebloede1070]

I agree with you

[@gavindeane3670]

​@@robertsmith4466Pi is not 22/7. Who told you that??? 22/7 is a common approximation of pi. It is not the exact value of pi. As I said in the previous comment, pi is an irrational number. It cannot be represented exactly in fraction or decimal form.

[@oldrrocr]

making me hungry doing this... because in grad school we calculated the relative advantage* of getting a large half-and-half pizza vs two small pizzas.🤣🤣🤣

[@melindaflynn2110]

2r

[@tomtke7351]

area of circle = pi×r^2 two concentric circles: C.1: r.1= 12 C.2: r.2= 6 Ring area: A.R = A.C1 - A.C2 A.C1 = pi × r.1^2 A.C2 = pi × r.2^2 A.R = pi × r.1^2 - pi × r.2^2 = pi (r.1^2 - r.2^2) = pi(12^2-6^2) = pi(144-36) = pi(108)cm^2 = 108pi cm^2

[@panlomito]

In that case I would say r1 = 12 = 2 . r2 so A1 = pi . (2 . r2)² = pi . 4 . r2² and A2 = pi . r2² A ring = A1 - A2 = pi . r2² . (4 - 1) = pi . r2² . 3 r2 = 6 cm so A ring = pi . 6² . 3 = 108 . pi cm² or r2 = 6 = ½ . r1 so A1 = pi . r1² and A2 = pi . (½ . r1)² = pi . ¼ . r1² A ring = A1 - A2 = pi . r1² . (1 - ¼) = pi . r1² . ¾ r1 = 12 cm so A ring = pi . 12² . ¾ = pi . 144 . 3 / 4 = pi . 36 . 3 = 108 . pi cm²

[@Budgeman83030]

I’m jumping ahead here but in calculus, wouldn’t this be solved using the washer method?

[@karlrschneider]

Your small 'rdius' looks like a diameter so I ended up with 135 * Pi. Grr. The units are immaterial.