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Therefore, the radius is half the diameter."},{"text":"$$r = 2d$","correct":false,"explanation":"This reverses and enlarges the relationship. The diameter is twice the radius, so rearranging $d = 2r$ correctly gives $r = \\frac{d}{2}$."},{"text":"$$r = \\frac{d}{4}$","correct":false,"explanation":"This incorrectly divides the diameter by four. A diameter consists of exactly two radii placed end to end, so the correct relationship is $r = \\frac{d}{2}$."},{"text":"$$r = d$","correct":false,"explanation":"This treats the radius and diameter as equal lengths. The diameter spans two radii through the centre, so the correct relationship is $r = \\frac{d}{2}$."}]},{"id":"417e13cd-155d-5937-1268-424385cfbbcd","cardType":"cloze","points":null,"front":"To avoid losing accuracy, rounding should only be performed [ ] of the calculation.","back":"To avoid losing accuracy, rounding should only be performed at the end of the calculation.","mcqOptions":[{"text":"at the end","correct":true,"explanation":"Keeping full calculator values throughout prevents small rounding errors from accumulating. Rounding only the final result gives the greatest possible accuracy for the required answer."},{"text":"at the start","correct":false,"explanation":"Rounding at the start introduces an error that affects every later step. Full-precision values should be retained, with rounding performed only at the end of the calculation."},{"text":"at each stage","correct":false,"explanation":"Repeated rounding allows errors to accumulate across several steps, potentially changing the final answer. Intermediate values should remain unrounded until the calculation is complete."},{"text":"halfway through","correct":false,"explanation":"Rounding an intermediate result reduces the accuracy of all calculations that follow it. Rounding should be delayed until the end, after the full-precision result has been found."}]},{"id":"4afe970d-e9f4-7bb2-2bab-cdf107e0a033","cardType":"basic","points":1,"front":"Which measurement of a cone is used to calculate its volume?","back":"The vertical height ($h$).","mcqOptions":[{"text":"The vertical height ($h$).","correct":true,"explanation":"Cone volume uses the perpendicular distance from the base to the vertex. This vertical height appears in $V = \\frac{1}{3}\\pi r^2h$."},{"text":"The slant height ($l$).","correct":false,"explanation":"The slant height measures along the cone’s curved surface and is used for curved surface area. Volume instead requires the vertical height $h$."},{"text":"The base diameter ($d$).","correct":false,"explanation":"The standard cone volume formula uses the base radius $r$, not the diameter as its height measurement. The required height is the vertical height $h$."},{"text":"The base circumference ($C$).","correct":false,"explanation":"Base circumference is not the height used in the cone volume formula. Volume requires the perpendicular vertical height $h$, together with the base radius."}]},{"id":"e591f4c9-25d0-e1de-0315-22b184c64d36","cardType":"basic","points":1,"front":"Which measurement of a cone is used to calculate its curved surface area?","back":"The slant height ($l$).","mcqOptions":[{"text":"The slant height ($l$).","correct":true,"explanation":"The curved surface follows the slope from the base rim to the vertex, so its area uses the slant height in the formula $\\pi rl$."},{"text":"The vertical height ($h$).","correct":false,"explanation":"Vertical height is used for cone volume because it is perpendicular to the base. Curved surface area follows the slope, so it requires the slant height $l$."},{"text":"The base diameter ($d$).","correct":false,"explanation":"The base diameter can be converted into a radius, but it is not the cone’s required height measurement. Curved surface area uses the slant height $l$."},{"text":"The base circumference ($C$).","correct":false,"explanation":"The base circumference does not describe the length of the cone’s curved slope. The required measurement for curved surface area is the slant height $l$."}]},{"id":"58cfe355-b53a-d49e-69a2-e3a3f5637c7e","cardType":"cloze","points":null,"front":"The volume of a cone is calculated using the formula $V = \\text{[...]}$.","back":"The volume of a cone is calculated using the formula $V = \\boxed{\\frac{1}{3}\\pi r^2 h}$.","mcqOptions":[{"text":"$$\\frac{1}{3}\\pi r^2 h$","correct":true,"explanation":"A cone has one-third the volume of a cylinder with the same circular base and vertical height. Therefore, its volume is $\\frac{1}{3}\\pi r^2h$."},{"text":"$$\\pi r^2 h$","correct":false,"explanation":"This is the volume formula for a cylinder, not a cone. A cone occupies one-third of the corresponding cylinder, giving $V = \\frac{1}{3}\\pi r^2h$."},{"text":"$$\\frac{1}{2}\\pi r^2 h$","correct":false,"explanation":"Using one-half confuses the cone’s volume factor with formulas involving triangular cross-sections. A cone has one-third of the matching cylinder’s volume, so the factor is $\\frac{1}{3}$."},{"text":"$$\\frac{1}{3}\\pi r h$","correct":false,"explanation":"This fails to square the radius when finding the circular base area. Since the base area is $\\pi r^2$, the volume is $\\frac{1}{3}\\pi r^2h$."}]},{"id":"48f35d0e-b078-8971-6628-84680364a6b9","cardType":"cloze","points":null,"front":"The volume of a sphere is calculated using the formula $V = \\text{[...]}$.","back":"The volume of a sphere is calculated using the formula $V = \\boxed{\\frac{4}{3}\\pi r^3}$.","mcqOptions":[{"text":"$$\\frac{4}{3}\\pi r^3$","correct":true,"explanation":"The volume of a sphere depends on the cube of its radius because volume is three-dimensional. Its exact formula is $V = \\frac{4}{3}\\pi r^3$."},{"text":"$$4\\pi r^2$","correct":false,"explanation":"This is the surface area of a sphere and uses square units, not cubic units. Sphere volume is calculated using $V = \\frac{4}{3}\\pi r^3$."},{"text":"$$\\frac{2}{3}\\pi r^3$","correct":false,"explanation":"This gives half the required volume and is the formula for a hemisphere. A complete sphere has volume $V = \\frac{4}{3}\\pi r^3$."},{"text":"$$\\frac{3}{4}\\pi r^3$","correct":false,"explanation":"This reverses the fraction in the sphere volume formula. The correct coefficient is $\\frac{4}{3}$, so the volume is $V = \\frac{4}{3}\\pi r^3$."}]},{"id":"48ea2515-0b80-cf46-d23e-f34071d608ba","cardType":"basic","points":1,"front":"What is the volume of a hemisphere in terms of $r$?","back":"$$\\frac{2}{3}\\pi r^3$","mcqOptions":[{"text":"$$\\frac{2}{3}\\pi r^3$","correct":true,"explanation":"A hemisphere is exactly half a sphere. Halving the sphere volume $\\frac{4}{3}\\pi r^3$ gives $\\frac{2}{3}\\pi r^3$."},{"text":"$$\\frac{4}{3}\\pi r^3$","correct":false,"explanation":"This is the volume of a complete sphere. A hemisphere is half of that volume, so its volume is $\\frac{2}{3}\\pi r^3$."},{"text":"$$\\frac{1}{3}\\pi r^3$","correct":false,"explanation":"This incorrectly quarters the full sphere formula rather than halving it. A hemisphere is half a sphere, giving $\\frac{2}{3}\\pi r^3$."},{"text":"$$2\\pi r^3$","correct":false,"explanation":"This multiplies the coefficient incorrectly when halving the sphere’s volume. Half of $\\frac{4}{3}$ is $\\frac{2}{3}$, so the result is $\\frac{2}{3}\\pi r^3$."}]},{"id":"4d1ef0fd-9cbe-0490-a0d8-0e7d3bad2fed","cardType":"basic","points":1,"front":"What does the formula $\\pi rl$ represent for a cone?","back":"The curved surface area.","mcqOptions":[{"text":"The curved surface area.","correct":true,"explanation":"For a cone, multiplying $\\pi$, the base radius $r$, and the slant height $l$ gives the area of the curved surface only."},{"text":"The total surface area.","correct":false,"explanation":"Total surface area must also include the circular base area $\\pi r^2$. Therefore, $\\pi rl$ alone represents only the curved surface area."},{"text":"The circular base area.","correct":false,"explanation":"The circular base has area $\\pi r^2$ and does not involve the slant height. The formula $\\pi rl$ represents the cone’s curved surface area."},{"text":"The volume of the cone.","correct":false,"explanation":"Volume uses the vertical height and has formula $\\frac{1}{3}\\pi r^2h$. The expression $\\pi rl$ instead calculates the cone’s curved surface area."}]},{"id":"8716da4a-2f65-9dbe-c7e8-76bc5c5bc4d8","cardType":"basic","points":2,"front":"How is the total surface area of a solid cone calculated?","back":"

Curved surface area ($\\pi rl$)
Base area ($\\pi r^2$)

","mcqOptions":[{"text":"

Curved surface area ($\\pi rl$)
Base area ($\\pi r^2$)

","correct":true,"explanation":"A solid cone has a curved surface and a circular base. Adding $\\pi rl$ and $\\pi r^2$ gives its total surface area."},{"text":"

Curved surface area ($\\pi r^2$)
Base area ($\\pi rl$)

","correct":false,"explanation":"This swaps the two formulae. The curved surface area is $\\pi rl$, while the circular base has area $\\pi r^2$; these must be added."},{"text":"

Curved surface area ($2\\pi rl$)
Base area ($\\pi r^2$)

","correct":false,"explanation":"The curved surface has been incorrectly doubled. A cone's curved surface area is $\\pi rl$, and adding the base area $\\pi r^2$ gives the total."},{"text":"

Curved surface area ($\\pi rl$)
Base area ($2\\pi r^2$)

","correct":false,"explanation":"The circular base has been counted twice. There is one base with area $\\pi r^2$, which is added to the curved surface area $\\pi rl$."}]},{"id":"609da800-75d0-08f1-e5b5-c23206e02062","cardType":"basic","points":1,"front":"What is the surface area of a sphere?","back":"$$4\\pi r^2$","mcqOptions":[{"text":"$$4\\pi r^2$","correct":true,"explanation":"The surface area of a sphere is four times the area of a circle with the same radius, so the formula is $4\\pi r^2$."},{"text":"$$\\pi r^2$","correct":false,"explanation":"This is the area of a single circle, not the surface area of a sphere. A sphere's complete surface area is $4\\pi r^2$."},{"text":"$$2\\pi r^2$","correct":false,"explanation":"This is the curved surface area of a hemisphere. A complete sphere has twice that curved area, giving $4\\pi r^2$."},{"text":"$$4\\pi r^3$","correct":false,"explanation":"Using $r^3$ produces cubic units, which measure volume rather than area. Surface area requires square units, and the correct formula is $4\\pi r^2$."}]},{"id":"17e5cd8e-7b9c-204a-4ef2-2a7aa8b06025","cardType":"cloze","points":null,"front":"The total surface area of a solid hemisphere is $\\text{[...]}$.","back":"The total surface area of a solid hemisphere is $\\boxed{3\\pi r^2}$.","mcqOptions":[{"text":"$$3\\pi r^2$","correct":true,"explanation":"A solid hemisphere includes curved area $2\\pi r^2$ and a circular base of area $\\pi r^2$. Their sum is $3\\pi r^2$."},{"text":"$$2\\pi r^2$","correct":false,"explanation":"This includes only the hemisphere's curved surface and omits its flat circular base. Adding the base area $\\pi r^2$ gives $3\\pi r^2$."},{"text":"$$4\\pi r^2$","correct":false,"explanation":"This is the surface area of a complete sphere. A solid hemisphere has half that curved area plus one circular base, giving $3\\pi r^2$."},{"text":"$$\\frac{4}{3}\\pi r^2$","correct":false,"explanation":"The coefficient $\\frac{4}{3}$ is associated with the sphere volume formula, not hemisphere surface area. Adding curved and base areas gives $3\\pi r^2$."}]},{"id":"4c3f8ad3-8fde-4f16-7903-c81b7d26938f","cardType":"basic","points":2,"front":"Why does a solid hemisphere have a total surface area of $3\\pi r^2$?","back":"It combines the curved surface ($2\\pi r^2$) and the flat circular base ($\\pi r^2$).","mcqOptions":[{"text":"It combines the curved surface ($2\\pi r^2$) and the flat circular base ($\\pi r^2$).","correct":true,"explanation":"Half a sphere has curved area $2\\pi r^2$, and its exposed circular base has area $\\pi r^2$. Together these total $3\\pi r^2$."},{"text":"It combines the curved surface ($\\pi r^2$) and the flat circular base ($2\\pi r^2$).","correct":false,"explanation":"The component areas have been reversed. The curved hemisphere contributes $2\\pi r^2$, while its single flat circular base contributes $\\pi r^2$."},{"text":"It combines the curved surface ($4\\pi r^2$) and the flat circular base ($\\pi r^2$).","correct":false,"explanation":"The curved area shown is for a complete sphere, not a hemisphere. Halving it gives $2\\pi r^2$, then the base adds $\\pi r^2$."},{"text":"It combines the curved surface ($2\\pi r^2$) and the flat circular base ($2\\pi r^2$).","correct":false,"explanation":"The flat base has been doubled. It is one circle with area $\\pi r^2$, so adding it to $2\\pi r^2$ gives $3\\pi r^2$."}]},{"id":"cfa3732a-cf5f-bf89-0177-aaba745bd022","cardType":"cloze","points":null,"front":"In a cone, the relationship between slant height ($l$), vertical height ($h$), and radius ($r$) is $\\text{[...]}$.","back":"In a cone, the relationship between slant height ($l$), vertical height ($h$), and radius ($r$) is $\\boxed{l^2 = h^2 + r^2}$.","mcqOptions":[{"text":"$$l^2 = h^2 + r^2$","correct":true,"explanation":"The vertical height and radius are perpendicular sides of a right-angled triangle, while the slant height is its hypotenuse. Pythagoras therefore gives $l^2=h^2+r^2$."},{"text":"$$h^2 = l^2 + r^2$","correct":false,"explanation":"This incorrectly treats the vertical height as the hypotenuse. The slant height is opposite the right angle, so the correct relationship is $l^2=h^2+r^2$."},{"text":"$$r^2 = h^2 + l^2$","correct":false,"explanation":"This incorrectly treats the radius as the hypotenuse. The slant height is the longest side of the right-angled triangle, giving $l^2=h^2+r^2$."},{"text":"$$l = h + r$","correct":false,"explanation":"Pythagoras requires the squares of the perpendicular sides to be added, not their lengths directly. The correct relationship is $l^2=h^2+r^2$."}]},{"id":"753c1d70-6a6b-8160-c7e7-4b01749232c5","cardType":"basic","points":1,"front":"What theorem is used to find the slant height of a cone if only the vertical height and radius are known?","back":"Pythagoras' theorem.","mcqOptions":[{"text":"Pythagoras' theorem.","correct":true,"explanation":"A cone's vertical height and radius meet at a right angle, with the slant height forming the hypotenuse. Pythagoras' theorem therefore finds the slant height."},{"text":"Cosine rule.","correct":false,"explanation":"The cosine rule is mainly needed for non-right-angled triangles or when an included angle is known. Here, the right angle makes Pythagoras' theorem the direct method."},{"text":"Sine rule.","correct":false,"explanation":"The sine rule requires known angles and opposite sides, which are not provided here. The height and radius are perpendicular, so Pythagoras' theorem applies."},{"text":"Circle theorem.","correct":false,"explanation":"Circle theorems determine angles in circles and do not relate the three side lengths here. The cone's cross-section forms a right-angled triangle, so use Pythagoras' theorem."}]},{"id":"83c29289-1f89-69f0-3f95-eb64fba9891e","cardType":"basic","points":1,"front":"How do you calculate the total volume of a compound solid?","back":"Add the volumes of the individual shapes together.","mcqOptions":[{"text":"Add the volumes of the individual shapes together.","correct":true,"explanation":"A compound solid is formed from multiple component solids. Calculate each component's volume using its appropriate formula, then add those volumes to obtain the total."},{"text":"Subtract the volumes of the individual shapes together.","correct":false,"explanation":"Subtraction is used only when a section is removed or hollowed out. For joined component shapes, add their individual volumes to find the total volume."},{"text":"Multiply the volumes of the individual shapes together.","correct":false,"explanation":"Multiplying volumes produces incorrect units and does not measure the combined space occupied. The total volume is found by adding the component volumes."},{"text":"Average the volumes of the individual shapes together.","correct":false,"explanation":"An average gives a typical component volume, not the space occupied by the complete solid. Add all the individual volumes to find the total."}]},{"id":"b6eaedbe-6afa-3154-a113-7da8a77e75d1","cardType":"basic","points":1,"front":"What does it mean to give an answer \"in terms of $\\pi$\"?","back":"Leave the symbol $\\pi$ in the final answer instead of converting to a decimal.","mcqOptions":[{"text":"Leave the symbol $\\pi$ in the final answer instead of converting to a decimal.","correct":true,"explanation":"An answer in terms of $\\pi$ keeps $\\pi$ as an exact symbol, usually multiplied by a number, rather than replacing it with a rounded decimal value."},{"text":"Replace the symbol $\\pi$ with $3.14$ and give the final answer as a decimal.","correct":false,"explanation":"Using $3.14$ creates a decimal approximation and removes $\\pi$. An answer in terms of $\\pi$ must retain the symbol in the final exact answer."},{"text":"Use the symbol $\\pi$ while working, then convert the final answer to a decimal.","correct":false,"explanation":"Converting at the end still produces a decimal approximation rather than an answer in terms of $\\pi$. The final answer itself must keep the $\\pi$ symbol."},{"text":"Replace the symbol $\\pi$ with the calculator value and round the final answer.","correct":false,"explanation":"Using the calculator value produces a rounded numerical approximation. To answer in terms of $\\pi$, leave $\\pi$ unchanged in the exact final expression."}]},{"id":"d7991ace-263b-4e9b-3ee1-6c457b937b44","cardType":"cloze","points":null,"front":"When a solid is fully submerged, the rise in water level = $\\text{[...]}$.","back":"When a solid is fully submerged, the rise in water level = $\\boxed{\\frac{\\text{volume of solid}}{\\text{area of container base}}}$.","mcqOptions":[{"text":"$$\\frac{\\text{volume of solid}}{\\text{area of container base}}$","correct":true,"explanation":"The displaced volume equals the solid’s volume. Since a prism-shaped container has volume equal to base area multiplied by height, the rise is volume divided by base area."},{"text":"$$\\frac{\\text{area of container base}}{\\text{volume of solid}}$","correct":false,"explanation":"This reverses the required division and does not produce a measurement of length. The rise is the volume of the solid divided by the area of the container base."},{"text":"$$\\frac{\\text{surface area of solid}}{\\text{area of container base}}$","correct":false,"explanation":"This incorrectly uses the solid’s surface area rather than its displaced volume. The water-level rise equals the volume of the solid divided by the container’s base area."},{"text":"$$\\frac{\\text{volume of solid}}{\\text{perimeter of container base}}$","correct":false,"explanation":"Dividing by the base perimeter is a common confusion between perimeter and area. The rise is found by dividing the solid’s volume by the container’s base area."}]},{"id":"9049225f-5476-308c-5b6b-4bd14c2f2d6c","cardType":"basic","points":1,"front":"How does the volume of a submerged solid relate to the water it displaces?","back":"The volume of the solid equals the volume of the water displaced.","mcqOptions":[{"text":"The volume of the solid equals the volume of the water displaced.","correct":true,"explanation":"A fully submerged solid takes up space that the water previously occupied, so it displaces a volume of water exactly equal to its own volume."},{"text":"The volume of the solid is twice the volume of the water displaced.","correct":false,"explanation":"Submersion does not introduce a factor of two. A fully submerged solid displaces a volume of water exactly equal to the volume of the solid."},{"text":"The volume of the solid is half the volume of the water displaced.","correct":false,"explanation":"Halving the displaced volume confuses full submersion with partial submersion. When the solid is fully submerged, its volume equals the volume of water displaced."},{"text":"The volume of the solid exceeds the volume of the water displaced.","correct":false,"explanation":"A fully submerged object cannot occupy more volume than the water it displaces. Its entire volume replaces an equal volume of water, regardless of its density."}]},{"id":"2950688b-52e1-5dcf-87ba-e18d8ea8464d","cardType":"basic","points":1,"front":"What operation is used to find the radius of a sphere if the volume is known?","back":"A cube root.","mcqOptions":[{"text":"A cube root.","correct":true,"explanation":"The sphere formula contains $r^3$, so after rearranging to isolate $r^3$, taking the cube root gives the radius."},{"text":"A square root.","correct":false,"explanation":"A square root would undo $r^2$, which appears in area formulas. Sphere volume contains $r^3$, so the radius is found using a cube root."},{"text":"A division by three.","correct":false,"explanation":"Dividing by three does not undo a power of three. After rearranging the sphere’s volume formula, a cube root is needed to recover the radius."},{"text":"A multiplication by three.","correct":false,"explanation":"Multiplying by three confuses an exponent with a coefficient. Since the volume formula contains the radius cubed, the final operation is a cube root."}]},{"id":"f9145fcd-9ab9-9ee6-2aac-1372c86fe177","cardType":"cloze","points":null,"front":"In cone formulas, $l$ represents the [ ] and $h$ represents the vertical height.","back":"In cone formulas, $l$ represents the slant height and $h$ represents the vertical height.","mcqOptions":[{"text":"slant height","correct":true,"explanation":"In cone formulas, $l$ is the distance along the cone’s sloping surface from the rim to the apex, so it represents the slant height."},{"text":"vertical height","correct":false,"explanation":"The vertical height is represented by $h$, not $l$. The symbol $l$ denotes the slant height measured along the cone’s sloping side."},{"text":"base radius","correct":false,"explanation":"The base radius is normally represented by $r$. In cone formulas, $l$ denotes the slant height from the edge of the circular base to the apex."},{"text":"base diameter","correct":false,"explanation":"The base diameter is twice the radius and is not represented by $l$. The symbol $l$ denotes the cone’s slant height."}]},{"id":"429c38d9-8487-33b2-3d82-541f8be54b7a","cardType":"cloze","points":null,"front":"In cone formulas, $l$ represents the slant height and $h$ represents the [ ].","back":"In cone formulas, $l$ represents the slant height and $h$ represents the vertical height.","mcqOptions":[{"text":"vertical height","correct":true,"explanation":"In cone formulas, $h$ is the perpendicular distance from the centre of the circular base to the apex, so it represents the vertical height."},{"text":"slant height","correct":false,"explanation":"The slant height is represented by $l$, not $h$. The symbol $h$ denotes the perpendicular vertical height from the base to the apex."},{"text":"base radius","correct":false,"explanation":"The base radius is represented by $r$, while $h$ denotes the cone’s vertical height measured perpendicularly from the base to the apex."},{"text":"base diameter","correct":false,"explanation":"The base diameter is twice the radius and is not represented by $h$. In cone formulas, $h$ denotes the perpendicular vertical height."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Unit 1"},"heading":"Spheres and Cones","title":"Spheres and Cones Flashcards","topbar":{"backHref":"/gcse/maths/wjec/flashcards"},"topicSidebar":"$L37"}]}],"topicId":"c19b7e75-4787-e2db-6cf6-c290e74e6d1f","walled":"$L38"}],"$L39","$L3a"]