GCSE Maths: Spheres and hemispheres (Higher)
Part of the GCSE Maths volume, surface area and 3d shapes chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Spheres and hemispheres
A sphere is a perfectly round solid, like a football or a marble. Every point on its surface is the same distance, the radius r, from the centre. A hemisphere is exactly half a sphere, for example a bowl or a dome. Two formulae describe a sphere, and both are printed on the Edexcel Higher formula sheet:
Volume of a sphere = &frac43;πr3 Surface area of a sphere = 4πr2
Notice that volume uses r3 (three lengths multiplied, cubic units) and surface area uses r2 (two lengths, square units). The surface area is exactly four times the area of the widest circle of the sphere.
Hemispheres. Halve everything for the volume and the curved surface:
Volume = ⅔πr3 Curved surface area = 2πr2 Total surface area = 2πr2 + πr2 = 3πr2
The extra πr2 in the total surface area is the flat circular face. It is only included if the hemisphere is solid (a dome sitting on the ground counts the flat face as part of the surface of the solid; an open bowl does not have it).
Working backwards. If you are given the volume or surface area and must find the radius, write the formula with the numbers, cancel π, then undo the operations in reverse: for volume you finish with a cube root, for surface area with a square root.
Key ideas
- Sphere: V = &frac43;πr3, surface area = 4πr2.
- Hemisphere: V = ⅔πr3, curved surface = 2πr2, total surface = 3πr2 (including the flat face).
- Use the radius: halve the diameter first.
- “In terms of π”: leave π as a symbol.
- To find r from a volume: cancel π, rearrange, then cube root. From a surface area: cancel π, divide by 4 (or 2 or 3), then square root.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Volume of a sphere = &frac43;πr3.
- Write down the formula. This is the volume of a sphere (it is on the formula sheet).V = &frac43;πr3
- Substitute. V = &frac43; × π × 33
- Cube the radius. r3 means r × r × r.33 = 3 × 3 × 3 = 27
- Multiply by &frac43;. Multiply by 4, then divide by 3 (or divide by 3 first if it is easier).27 × 4 = 108, then ÷ 3 = 36
- Write with π. V = 36π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.36π cm3
Check. Estimate: a sphere fills about half of the cube that just holds it. That cube has volume 216, half is 108, and our answer 113 is a little more than that. ✓
Watch out. r3 means r × r × r, not r × 3. Use the radius: half the diameter.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Surface area of a sphere = 4πr2.
- Find the radius. The formulas use the radius, which is half the diameter.r = 10 ÷ 2 = 5
- Write down the formula. The surface area of a sphere is exactly four times the area of its widest circle.Surface area = 4πr2
- Substitute and square the radius first. Powers before multiplying.4 × π × 52 = 4 × π × 25
- Multiply. 4 × 25 = 100, so the area is 100π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.4 × π × 52 = 314.1593…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.314.1593… = 314 cm2
Check. It is 4 times the area of a circle with the same radius: π × 52 = 25π, and 4 × 25π = 100π. ✓
Watch out. Square the radius first, then multiply by 4π. Do not square the 4.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Volume of a hemisphere = half of &frac43;πr3 = ⅔πr3.
- Write the sphere formula. A hemisphere is half of a sphere, so its volume is half of the sphere volume.Sphere: V = &frac43;πr3
- Halve it. ½ of &frac43; is ⅔.Hemisphere: V = ½ × &frac43;πr3 = ⅔πr3
- Substitute and cube the radius. V = ⅔ × π × 63, and 63 = 216
- Multiply by ⅔. Multiply by 2, then divide by 3.216 × 2 = 432, ÷ 3 = 144, so V = 144π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.144π cm3
Check. Half the sphere volume: 288π ÷ 2 = 144π. ✓
Watch out. A hemisphere has half the volume of the full sphere: halve the &frac43;, do not halve the radius.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Total surface area of a solid hemisphere = curved surface (2πr2) + flat circle (πr2) = 3πr2.
- Curved surface. Half of the sphere surface area 4πr2.2πr2 = 2 × π × 32 = 18π
- Flat circular face. A solid hemisphere has a flat circle on the bottom. Area of a circle = πr2.π × 32 = 9π
- Add them. Add the numbers in front of π.18π + 9π = 27π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.27π cm2
Check. 3πr2 = 3 × π × 9 = 27π. ✓
Watch out. Do not forget the flat circular face of a solid hemisphere: 2πr2 + πr2 = 3πr2.
Method. Put the volume into V = &frac43;πr3, cancel π, get r3 on its own, then cube root.
- Write the formula with the numbers. The volume is given as a multiple of π. The radius r is unknown.36π = &frac43; × π × r3
- Divide both sides by π. The π cancels.36 = &frac43; × r3
- Multiply by 3 and divide by 4. This undoes the &frac43;. (Multiply by the reciprocal ¾.)r3 = 36 × 3 ÷ 4 = 108 ÷ 4 = 27
- Cube root. Find the number that multiplies by itself three times to give r3.r = ∛27 = 3 cm (because 3 × 3 × 3 = 27)
Check. &frac43; × π × 33 = &frac43; × π × 27 = 36π. ✓
Watch out. Undo &frac43; by multiplying by ¾. Then take a cube root, not a square root.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Volume of a sphere = &frac43;πr3.
- Find the radius. The formulas use the radius, which is half the diameter.r = 6.7 ÷ 2 = 3.35
- Write down the formula. This is the volume of a sphere (it is on the formula sheet).V = &frac43;πr3
- Substitute. V = &frac43; × π × 3.353
- Cube the radius. r3 means r × r × r.3.353 = 3.35 × 3.35 × 3.35 = 37.595375
- Multiply by &frac43;. Multiply by 4, then divide by 3 (or divide by 3 first if it is easier).37.595375 × 4 = 150.3815, then ÷ 3 = 507636000
- Write with π. V = 3007636000π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.&frac43; × π × 3.353 = 157.4791…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.157.4791… = 157 cm3
Check. Estimate: a sphere fills about half of the cube that just holds it. That cube has volume 300.763, half is 150.3815, and our answer 157 is a little more than that. ✓
Watch out. r3 means r × r × r, not r × 3. Use the radius: half the diameter.
Watch out
Cube the radius for volume (r3) and square it for surface area (r2). Do not confuse the formulae. If you are given the diameter, halve it first. A solid hemisphere has a flat face, so its total surface area is 3πr2, not 2πr2.Exercise 19K Spheres and hemispheres
Non-calculatorKey words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Volume of a sphere = &frac43;πr3.
- Write down the formula. This is the volume of a sphere (it is on the formula sheet).V = &frac43;πr3
- Substitute. V = &frac43; × π × 33
- Cube the radius. r3 means r × r × r.33 = 3 × 3 × 3 = 27
- Multiply by &frac43;. Multiply by 4, then divide by 3 (or divide by 3 first if it is easier).27 × 4 = 108, then ÷ 3 = 36
- Write with π. V = 36π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.36π cm3
Check. Estimate: a sphere fills about half of the cube that just holds it. That cube has volume 216, half is 108, and our answer 113 is a little more than that. ✓
Watch out. r3 means r × r × r, not r × 3. Use the radius: half the diameter.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Volume of a sphere = &frac43;πr3.
- Write down the formula. This is the volume of a sphere (it is on the formula sheet).V = &frac43;πr3
- Substitute. V = &frac43; × π × 63
- Cube the radius. r3 means r × r × r.63 = 6 × 6 × 6 = 216
- Multiply by &frac43;. Multiply by 4, then divide by 3 (or divide by 3 first if it is easier).216 × 4 = 864, then ÷ 3 = 288
- Write with π. V = 288π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.288π cm3
Check. Estimate: a sphere fills about half of the cube that just holds it. That cube has volume 1728, half is 864, and our answer 905 is a little more than that. ✓
Watch out. r3 means r × r × r, not r × 3. Use the radius: half the diameter.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Volume of a sphere = &frac43;πr3.
- Find the radius. The formulas use the radius, which is half the diameter.r = 10 ÷ 2 = 5
- Write down the formula. This is the volume of a sphere (it is on the formula sheet).V = &frac43;πr3
- Substitute. V = &frac43; × π × 53
- Cube the radius. r3 means r × r × r.53 = 5 × 5 × 5 = 125
- Multiply by &frac43;. Multiply by 4, then divide by 3 (or divide by 3 first if it is easier).125 × 4 = 500, then ÷ 3 = 16623
- Write with π. V = 5003π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.5003π cm3
Check. Estimate: a sphere fills about half of the cube that just holds it. That cube has volume 1000, half is 500, and our answer 524 is a little more than that. ✓
Watch out. r3 means r × r × r, not r × 3. Use the radius: half the diameter.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Volume of a sphere = &frac43;πr3.
- Write down the formula. This is the volume of a sphere (it is on the formula sheet).V = &frac43;πr3
- Substitute. V = &frac43; × π × 1.53
- Cube the radius. r3 means r × r × r.1.53 = 1.5 × 1.5 × 1.5 = 3.375
- Multiply by &frac43;. Multiply by 4, then divide by 3 (or divide by 3 first if it is easier).3.375 × 4 = 13.5, then ÷ 3 = 4.5
- Write with π. V = 4.5π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.4.5π cm3
Check. Estimate: a sphere fills about half of the cube that just holds it. That cube has volume 27, half is 13.5, and our answer 14 is a little more than that. ✓
Watch out. r3 means r × r × r, not r × 3. Use the radius: half the diameter.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Surface area of a sphere = 4πr2.
- Write down the formula. The surface area of a sphere is exactly four times the area of its widest circle.Surface area = 4πr2
- Substitute and square the radius first. Powers before multiplying.4 × π × 52 = 4 × π × 25
- Multiply. 4 × 25 = 100, so the area is 100π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.100π cm2
Check. It is 4 times the area of a circle with the same radius: π × 52 = 25π, and 4 × 25π = 100π. ✓
Watch out. Square the radius first, then multiply by 4π. Do not square the 4.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Surface area of a sphere = 4πr2.
- Write down the formula. The surface area of a sphere is exactly four times the area of its widest circle.Surface area = 4πr2
- Substitute and square the radius first. Powers before multiplying.4 × π × 72 = 4 × π × 49
- Multiply. 4 × 49 = 196, so the area is 196π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.196π cm2
Check. It is 4 times the area of a circle with the same radius: π × 72 = 49π, and 4 × 49π = 196π. ✓
Watch out. Square the radius first, then multiply by 4π. Do not square the 4.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Surface area of a sphere = 4πr2.
- Find the radius. The formulas use the radius, which is half the diameter.r = 12 ÷ 2 = 6
- Write down the formula. The surface area of a sphere is exactly four times the area of its widest circle.Surface area = 4πr2
- Substitute and square the radius first. Powers before multiplying.4 × π × 62 = 4 × π × 36
- Multiply. 4 × 36 = 144, so the area is 144π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.144π cm2
Check. It is 4 times the area of a circle with the same radius: π × 62 = 36π, and 4 × 36π = 144π. ✓
Watch out. Square the radius first, then multiply by 4π. Do not square the 4.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Volume of a hemisphere = half of &frac43;πr3 = ⅔πr3.
- Write the sphere formula. A hemisphere is half of a sphere, so its volume is half of the sphere volume.Sphere: V = &frac43;πr3
- Halve it. ½ of &frac43; is ⅔.Hemisphere: V = ½ × &frac43;πr3 = ⅔πr3
- Substitute and cube the radius. V = ⅔ × π × 33, and 33 = 27
- Multiply by ⅔. Multiply by 2, then divide by 3.27 × 2 = 54, ÷ 3 = 18, so V = 18π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.18π cm3
Check. Half the sphere volume: 36π ÷ 2 = 18π. ✓
Watch out. A hemisphere has half the volume of the full sphere: halve the &frac43;, do not halve the radius.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Volume of a hemisphere = half of &frac43;πr3 = ⅔πr3.
- Write the sphere formula. A hemisphere is half of a sphere, so its volume is half of the sphere volume.Sphere: V = &frac43;πr3
- Halve it. ½ of &frac43; is ⅔.Hemisphere: V = ½ × &frac43;πr3 = ⅔πr3
- Substitute and cube the radius. V = ⅔ × π × 63, and 63 = 216
- Multiply by ⅔. Multiply by 2, then divide by 3.216 × 2 = 432, ÷ 3 = 144, so V = 144π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.144π cm3
Check. Half the sphere volume: 288π ÷ 2 = 144π. ✓
Watch out. A hemisphere has half the volume of the full sphere: halve the &frac43;, do not halve the radius.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Volume of a hemisphere = half of &frac43;πr3 = ⅔πr3.
- Write the sphere formula. A hemisphere is half of a sphere, so its volume is half of the sphere volume.Sphere: V = &frac43;πr3
- Halve it. ½ of &frac43; is ⅔.Hemisphere: V = ½ × &frac43;πr3 = ⅔πr3
- Substitute and cube the radius. V = ⅔ × π × 93, and 93 = 729
- Multiply by ⅔. Multiply by 2, then divide by 3.729 × 2 = 1458, ÷ 3 = 486, so V = 486π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.486π cm3
Check. Half the sphere volume: 972π ÷ 2 = 486π. ✓
Watch out. A hemisphere has half the volume of the full sphere: halve the &frac43;, do not halve the radius.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Curved surface area of a hemisphere = half of 4πr2 = 2πr2.
- Write the sphere surface area. A hemisphere has half the curved surface of a sphere.Sphere: 4πr2
- Halve it. Curved surface = ½ × 4πr2 = 2πr2 = 2 × π × 52 = 50π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.50π cm2
Check. Twice this is 100π, the full sphere 4πr2 = 4 × π × 25 = 100π. ✓
Watch out. This is only the curved part. A solid hemisphere also has a flat circular face.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Total surface area of a solid hemisphere = curved surface (2πr2) + flat circle (πr2) = 3πr2.
- Curved surface. Half of the sphere surface area 4πr2.2πr2 = 2 × π × 52 = 50π
- Flat circular face. A solid hemisphere has a flat circle on the bottom. Area of a circle = πr2.π × 52 = 25π
- Add them. Add the numbers in front of π.50π + 25π = 75π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.75π cm2
Check. 3πr2 = 3 × π × 25 = 75π. ✓
Watch out. Do not forget the flat circular face of a solid hemisphere: 2πr2 + πr2 = 3πr2.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Volume of a sphere = &frac43;πr3.
- Write down the formula. This is the volume of a sphere (it is on the formula sheet).V = &frac43;πr3
- Substitute. V = &frac43; × π × 4.53
- Cube the radius. r3 means r × r × r.4.53 = 4.5 × 4.5 × 4.5 = 91.125
- Multiply by &frac43;. Multiply by 4, then divide by 3 (or divide by 3 first if it is easier).91.125 × 4 = 364.5, then ÷ 3 = 121.5
- Write with π. V = 121.5π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.&frac43; × π × 4.53 = 381.7035…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.381.7035… = 382 cm3
Check. Estimate: a sphere fills about half of the cube that just holds it. That cube has volume 729, half is 364.5, and our answer 382 is a little more than that. ✓
Watch out. r3 means r × r × r, not r × 3. Use the radius: half the diameter.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Volume of a sphere = &frac43;πr3.
- Find the radius. The formulas use the radius, which is half the diameter.r = 15 ÷ 2 = 7.5
- Write down the formula. This is the volume of a sphere (it is on the formula sheet).V = &frac43;πr3
- Substitute. V = &frac43; × π × 7.53
- Cube the radius. r3 means r × r × r.7.53 = 7.5 × 7.5 × 7.5 = 421.875
- Multiply by &frac43;. Multiply by 4, then divide by 3 (or divide by 3 first if it is easier).421.875 × 4 = 1687.5, then ÷ 3 = 562.5
- Write with π. V = 562.5π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.&frac43; × π × 7.53 = 1767.1459…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.1767.1459… = 1770 cm3
Check. Estimate: a sphere fills about half of the cube that just holds it. That cube has volume 3375, half is 1687.5, and our answer 1767 is a little more than that. ✓
Watch out. r3 means r × r × r, not r × 3. Use the radius: half the diameter.
Key words. A sphere is a perfectly round solid (like a ball). A hemisphere is half of a sphere. The radius is the same distance from the centre to any point on the surface.
Method. Surface area of a sphere = 4πr2.
- Find the radius. The formulas use the radius, which is half the diameter.r = 6.7 ÷ 2 = 3.35
- Write down the formula. The surface area of a sphere is exactly four times the area of its widest circle.Surface area = 4πr2
- Substitute and square the radius first. Powers before multiplying.4 × π × 3.352 = 4 × π × 11.2225
- Multiply. 4 × 11.2225 = 44.89, so the area is 44.89π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.4 × π × 3.352 = 141.0261…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.141.0261… = 141 cm2
Check. It is 4 times the area of a circle with the same radius: π × 3.352 = 11.2225π, and 4 × 11.2225π = 44.89π. ✓
Watch out. Square the radius first, then multiply by 4π. Do not square the 4.
Method. Put the volume into V = &frac43;πr3, cancel π, get r3 on its own, then cube root.
- Write the formula with the numbers. The volume is given as a multiple of π. The radius r is unknown.36π = &frac43; × π × r3
- Divide both sides by π. The π cancels.36 = &frac43; × r3
- Multiply by 3 and divide by 4. This undoes the &frac43;. (Multiply by the reciprocal ¾.)r3 = 36 × 3 ÷ 4 = 108 ÷ 4 = 27
- Cube root. Find the number that multiplies by itself three times to give r3.r = ∛27 = 3 cm (because 3 × 3 × 3 = 27)
Check. &frac43; × π × 33 = &frac43; × π × 27 = 36π. ✓
Watch out. Undo &frac43; by multiplying by ¾. Then take a cube root, not a square root.
Method. Put the volume into V = &frac43;πr3, cancel π, get r3 on its own, then cube root.
- Write the formula with the numbers. The volume is given as a multiple of π. The radius r is unknown.288π = &frac43; × π × r3
- Divide both sides by π. The π cancels.288 = &frac43; × r3
- Multiply by 3 and divide by 4. This undoes the &frac43;. (Multiply by the reciprocal ¾.)r3 = 288 × 3 ÷ 4 = 864 ÷ 4 = 216
- Cube root. Find the number that multiplies by itself three times to give r3.r = ∛216 = 6 cm (because 6 × 6 × 6 = 216)
Check. &frac43; × π × 63 = &frac43; × π × 216 = 288π. ✓
Watch out. Undo &frac43; by multiplying by ¾. Then take a cube root, not a square root.
Method. Put the volume into V = &frac43;πr3, cancel π, get r3 on its own, then cube root.
- Write the formula with the numbers. The volume is given as a multiple of π. The radius r is unknown.972π = &frac43; × π × r3
- Divide both sides by π. The π cancels.972 = &frac43; × r3
- Multiply by 3 and divide by 4. This undoes the &frac43;. (Multiply by the reciprocal ¾.)r3 = 972 × 3 ÷ 4 = 2916 ÷ 4 = 729
- Cube root. Find the number that multiplies by itself three times to give r3.r = ∛729 = 9 cm (because 9 × 9 × 9 = 729)
Check. &frac43; × π × 93 = &frac43; × π × 729 = 972π. ✓
Watch out. Undo &frac43; by multiplying by ¾. Then take a cube root, not a square root.
Method. Put the surface area into 4πr2, cancel π, divide by 4, then square root.
- Write the formula with the numbers. The surface area is given as a multiple of π.100π = 4 × π × r2
- Divide both sides by π. The π cancels.100 = 4 × r2
- Divide by 4. This leaves r2.r2 = 100 ÷ 4 = 25
- Square root. r = √25 = 5 cm
Check. 4 × π × 52 = 4 × π × 25 = 100π. ✓
Watch out. After cancelling π you must also divide by 4 before the square root.
Method. Put the surface area into 4πr2, cancel π, divide by 4, then square root.
- Write the formula with the numbers. The surface area is given as a multiple of π.144π = 4 × π × r2
- Divide both sides by π. The π cancels.144 = 4 × r2
- Divide by 4. This leaves r2.r2 = 144 ÷ 4 = 36
- Square root. r = √36 = 6 cm
Check. 4 × π × 62 = 4 × π × 36 = 144π. ✓
Watch out. After cancelling π you must also divide by 4 before the square root.
Method. Put the surface area into 4πr2, cancel π, divide by 4, then square root.
- Write the formula with the numbers. The surface area is given as a multiple of π.324π = 4 × π × r2
- Divide both sides by π. The π cancels.324 = 4 × r2
- Divide by 4. This leaves r2.r2 = 324 ÷ 4 = 81
- Square root. r = √81 = 9 cm
Check. 4 × π × 92 = 4 × π × 81 = 324π. ✓
Watch out. After cancelling π you must also divide by 4 before the square root.
Method. The bowl is half a sphere. Use the hemisphere volume, then change cm3 to litres.
- Write the hemisphere volume. Half of &frac43;πr3.V = ⅔πr3 = ⅔ × π × 123
- Cube 12. 12 × 12 = 144 and 144 × 12 = 1728.123 = 1728
- Multiply by ⅔. Divide 1728 by 3 to get 576, then multiply by 2.⅔ × 1728 = 1152, so V = 1152π
- Use the calculator. Use the π button.1152π = 3619.115… cm3
- Change to litres. 1 litre = 1000 cm3.3619.115… ÷ 1000 = 3.6191…
- Round to 3 significant figures. 3.62 litres
Check. A full sphere of radius 12 would be 4/3 × π × 1728 = 7238 cm3, and half of that is 3619 cm3. ✓
Method. The volume of water displaced equals the volume of the sphere. This volume also equals the base area of the tank × the rise in level.
- Volume of the sphere. V = &frac43;πr3 with r = 3.&frac43; × π × 27 = 36π cm3
- Write the volume of the raised water as a cylinder. It has radius 6 and unknown height h.π × 62 × h = 36πh
- Put the volumes equal. The water pushed up equals the sphere.36πh = 36π
- Solve. Divide both sides by 36π.h = 1 cm
Check. The sphere (36π) fills a layer of the tank of area 36π to depth 1. ✓
Method. Work out the volume of the sphere and of the cube and divide.
- Volume of the sphere. &frac43;πr3 with r = 5.&frac43; × π × 125 = 500π/3
- Volume of the cube. The side is the diameter, 10 cm.10 × 10 × 10 = 1000
- Divide the sphere by the cube. Write it as a fraction and simplify.(500π/3) ÷ 1000 = 500π ÷ 3000 = π ÷ 6 ✓
Check. π ÷ 6 is about 0.52, so the sphere fills just over half the cube. ✓
Method. Add the curved surface and the flat circular face.
- Curved surface. Half the surface area of a sphere, 4πr2 ÷ 2.2πr2
- Flat face. It is a circle of radius r.πr2
- Add. 2 lots of πr2 plus 1 lot.2πr2 + πr2 = 3πr2 ✓
Check. With r = 1: 2π + π = 3π. ✓
Method. Write both volumes in terms of r and divide.
- Volume of the cylinder. Substitute h = 2r into πr2h.V = πr2 × 2r = 2πr3
- Volume of the sphere. V = &frac43;πr3
- Divide. Cancel πr3 from both.&frac43; ÷ 2 = ⅔, so the sphere is two thirds of the cylinder. ✓
Check. Try r = 3: cylinder 54π, sphere 36π, and 36 ÷ 54 = ⅔. ✓
Method. Replace r by 2r in each formula and compare with the original.
- Surface area. The radius is squared in 4πr2.4π(2r)2 = 4π × 4r2 = 4 × (4πr2)
- Volume. The radius is cubed in &frac43;πr3.&frac43;π(2r)3 = &frac43;π × 8r3 = 8 × (&frac43;πr3)
- Conclude. Sam is not correct: the surface area is multiplied by 4 and the volume by 8.
Check. r = 1 and r = 2: areas 4π and 16π (4 times); volumes 4π/3 and 32π/3 (8 times). ✓
Method. Write the volume equal to the surface area and solve for r.
- Write the equation. The two formulae are equal numerically.&frac43;πr3 = 4πr2
- Divide both sides by 4πr2. This cancels π and two of the r factors. (r is not zero.)r ÷ 3 = 1
- Solve. Multiply both sides by 3.r = 3 cm
Check. r = 3: volume = 4/3 × π × 27 = 36π; surface area = 4 × π × 9 = 36π. ✓