Log in ARF Man 12 years agoPosted 12 years ago. Direct link to ARF Man's post “When Mr. Khan said the vo...” When Mr. Khan said the volume formula at • (83 votes) Josh Shaffer 12 years agoPosted 12 years ago. Direct link to Josh Shaffer's post “Since the Volume of a Sph...” Since the Volume of a Sphere is V=(4/3)πr^3, we are using the rational number 4/3 to give us the EXACT solution. Notice however that in this video, after Mr. Khan does his substitution, he uses his calculator to find an APPROXIMATE solution of V=(4/3)π(7)^3=1436.8. Had the problem specified to find the EXACT volume, then we would just substitute, V=(4/3)π(7)^3 and simplify to V=(4/3)π(343)=(1372/3)π, where we do NOT perform the division. (25 votes) Eric (хдх) 6 years agoPosted 6 years ago. Direct link to Eric (хдх)'s post “Where do you get 4/3 from...” Where do you get 4/3 from?! • (35 votes) Ian Pulizzotto 6 years agoPosted 6 years ago. Direct link to Ian Pulizzotto's post “Great question!! The 4/3...” Great question!! The 4/3 isn't so obvious and requires some work to derive. Consider the following two figures: From the volume formulas for a cylinder and a cone, the volume of Figure 2 is Now we need to compare the areas of the horizontal cross sections of Figure 1 and Figure 2 at any given height h above the bottom. Once we show that these cross sections have equal areas at every height, then Cavalieri's principle would imply that the volumes of Figure 1 and Figure 2 are equal (since the overall heights of the two figures are equal, specifically to r). In Figure 1, the cross section is a circle with radius sqrt(r^2 - h^2) from the Pythagorean Theorem (hypotenuse is r, one leg is h, and the other leg is the cross section's radius). In Figure 2, the cross section is a ring-shaped region with outer radius equal to r (from the cylinder, since each cross section's radius is the cylinder's radius) and inner radius equal to h (from the cone, since in a cone with equal height and radius, each cross section's radius equals its height above the bottom point). Therefore, these cross sections have equal areas at every height. So Figure 1 and Figure 2 have the same volume. (By the way, if you take calculus later, you will be able to derive this formula in another way by finding an integral. The volume of a full sphere is integral -r to r of pi(r^2 - x^2) dx. ) (90 votes) Natalie 7 years agoPosted 7 years ago. Direct link to Natalie's post “Is it easier to use 3.14 ...” Is it easier to use 3.14 while solving, or pi? My math teacher lets us do either, but I'm not sure which one would end up being more accurate. • (6 votes) kubleeka 7 years agoPosted 7 years ago. Direct link to kubleeka's post “Just using the symbol π i...” Just using the symbol π is infinitely more accurate than writing 3.14. The only catch is that leaving your answer in terms of π doesn't give you a decimal expansion of your answer. Experimental scientists and engineers will often use 3.14 (or even just 3), while mathematicians and more theoretical-focused people will use π. It just depends on what you want out of your answer. (27 votes) maggieolemiss 7 years agoPosted 7 years ago. Direct link to maggieolemiss's post “When I was doing my math ...” When I was doing my math homework, I was going along with the video and keying in the numbers on my page into the calculator. I ran into one problem. I didn't get the right answer and now I don't know what to do. My radius was 12, but my answer was 7238.229474. It's wrong though. I don't know what to do. Please help! • (8 votes) David Severin 7 years agoPosted 7 years ago. Direct link to David Severin's post “V = 4/3 π r^3, so first c...” V = 4/3 π r^3, so first check and see if the problem asks for the answer in terms of pi which would be 2304 π. If the question asks for the approximate answer, and we multiplied 2304 times π, your answer would be correct, so you should look at where they are asking you to round the number to, your rounding to the nearest millionth will almost always be overkill, the normal questions asking for rounding answers is to nearest whole number, nearest tenth, or nearest hundredth. So since the math is correct, then my assumption is that you did not answer the question to the accuracy that was asked (or in terms of pi). (20 votes) Lauren 4 years agoPosted 4 years ago. Direct link to Lauren's post “How do you find the surfa...” How do you find the surface area when you only have the volume? • (6 votes) Ian Pulizzotto 4 years agoPosted 4 years ago. Direct link to Ian Pulizzotto's post “I’m assuming you’re askin...” I’m assuming you’re asking about finding a sphere’s surface area, given its volume. Substitute the given volume for V in the equation V = (4/3)pi r^3 and solve for the radius r. Solving for r involves dividing both sides by (4/3)pi and then taking the cube root of both sides. Once you find r, substitute your value of r into the equation S = 4pi r^2 to find the surface area S. (13 votes) 💎Chυcκ Lørrε💎 6 years agoPosted 6 years ago. Direct link to 💎Chυcκ Lørrε💎's post “How do you deduce the for...” How do you deduce the formula? It's really important to me, please help! (I know it includes geometry when you deduce, because I need geometry when I'm deduce the formula of area of circles) • (8 votes) rut.fle.035 4 years agoPosted 4 years ago. Direct link to rut.fle.035's post “What is the formula for f...” What is the formula for finding the volume of a sphere with the same radius and height of a cylinder(vis- versa)? I have been searching the web and still have not found a clear answer. Please help me. Thank you! • (5 votes) David Severin 4 years agoPosted 4 years ago. Direct link to David Severin's post “You do need to be more sp...” You do need to be more specific, the volume of a sphere is V = 4/3 π r^3, it does not need to be related to a cylinder. So if you know the radius, you can calculate the volume. The volume of a cylinder with the same radius and with a height of 2r (since it would be the diameter across) would be V = π r^2 h = 2π r^3. So the empty space of a sphere placed in a cylinder would be V = 2πr^3 - 4/3πr^3 = 2/3 π r3. (5 votes) Nina Cilliers 8 months agoPosted 8 months ago. Direct link to Nina Cilliers's post “When figuring the volume ...” When figuring the volume out of a sphere, you're always going to have to multiply 4/3 with pi, right? But 4/3 times pi is like 4.1866666... infinity. So how should I round it off? Then surely my answer won't be exact anymore? Please help! (Also, I'm not allowed to use a calculator like he did in the video) • (5 votes) Isabella Herring 8 months agoPosted 8 months ago. Direct link to Isabella Herring's post “you could just use 3 of t...” you could just use 3 of the repeating number and ignore the rest of the digits no need to round up (3 votes) dominict20 4 years agoPosted 4 years ago. Direct link to dominict20's post “what is the volume of 1 a...” what is the volume of 1 and a half sphere • (4 votes) Ian Pulizzotto 4 years agoPosted 4 years ago. Direct link to Ian Pulizzotto's post “The volume of a sphere is...” The volume of a sphere is (4/3)pi r^3. So the total volume of a sphere and a hemisphere with the same radius is (3/2)(4/3)pi r^3 = 2pi r^3. (5 votes) MutanTGamr12609 a year agoPosted a year ago. Direct link to MutanTGamr12609's post “How do you find the surfa...” How do you find the surface area of a sphere? • (2 votes) David Severin a year agoPosted a year ago. Direct link to David Severin's post “the formula is 4π r^2.” the formula is 4π r^2. (8 votes)Want to join the conversation?
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Figure 1: the top half of a sphere with radius r.
Figure 2: a cylinder with radius r and height r, but with a cone (with point on bottom at the center of the cylinder's bottom base) with radius r and height r removed from it.
pi r^2 * r - (1/3) pi r^2 * r = (2/3) pi r^3.
So the area of the cross section at height h is pi[sqrt(r^2 - h^2)]^2 = pi(r^2 - h^2).
So the area of the cross section at height h is pi r^2 - pi h^2 = pi(r^2 - h^2).
Since we have found that the volume of Figure 2 is (2/3) pi r^3, the same is true for Figure 1, which is a hemisphere of radius r.
Therefore, the volume of a full sphere is (4/3) pi r^3.
See Also
Problem 25 Find the surface area of a cube ... [FREE SOLUTION]PowerSchool Learning Geometry Semester Wilson = 21.1 DOL Assessment Work Microsoft Edge https://dallasisd.learning.powerschool.com/tiowilson/geometrysemester2-wilson/assessments/take/866628742t=1619053007 No Time Limit Section
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