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circumference formula, which measures the distance around the circle. The area formula squares the radius, so the correct answer is $\\pi r^2$."},{"text":"$\\pi r$","correct":false,"explanation":"This omits the required square on the radius, so it does not calculate a two-dimensional area. The correct area formula is $\\pi r^2$."},{"text":"$\\pi r^3$","correct":false,"explanation":"Cubing the radius would produce cubic units, which are used for volume rather than area. A circle’s area is calculated using $\\pi r^2$."}]},{"id":"25fff780-c9c1-768f-60da-aa05ebe0f6fa","cardType":"basic","points":1,"front":"For a circle with radius $r$, what is the circumference formula?","back":"$2\\pi r$","mcqOptions":[{"text":"$2\\pi r$","correct":true,"explanation":"The circumference is the distance around a circle. Since the diameter is $2r$ and circumference is $\\pi$ times the diameter, the formula is $2\\pi r$."},{"text":"$\\pi r^2$","correct":false,"explanation":"This calculates the circle’s area rather than the distance around it. Circumference uses the radius without squaring it, giving the correct formula $2\\pi r$."},{"text":"$\\pi r$","correct":false,"explanation":"This misses the factor of $2$ caused by the diameter being twice the radius. The circumference formula is therefore $2\\pi r$."},{"text":"$2\\pi r^2$","correct":false,"explanation":"Squaring the radius incorrectly introduces square units, which measure area rather than length. The circumference is a length and is calculated using $2\\pi r$."}]},{"id":"ff363e8d-d279-d542-5452-438b7b9c5d04","cardType":"cloze","points":null,"front":"The diameter of a circle is [ ].","back":"The diameter of a circle is twice the radius.","mcqOptions":[{"text":"twice the radius","correct":true,"explanation":"The radius runs from the centre to the circumference, while the diameter crosses the full circle through the centre. It therefore consists of two radii."},{"text":"half the radius","correct":false,"explanation":"This reverses the relationship between radius and diameter. The radius is half the diameter, so the diameter is correctly described as twice the radius."},{"text":"equal to the radius","correct":false,"explanation":"This overlooks that a diameter extends from one side of the circle to the other through the centre. Its length is twice the radius."},{"text":"four times the radius","correct":false,"explanation":"A diameter contains two radii placed end to end, not four. Therefore, the diameter of a circle is twice the radius."}]},{"id":"3475f3ea-9a62-2860-6df8-0ca41b02539e","cardType":"cloze","points":null,"front":"An arc is part of the [ ] of a circle.","back":"An arc is part of the circumference of a circle.","mcqOptions":[{"text":"circumference","correct":true,"explanation":"An arc is a curved section of the boundary of a circle. Since that complete boundary is the circumference, an arc forms part of the circumference."},{"text":"radius","correct":false,"explanation":"A radius is a straight line from the centre to the circumference, whereas an arc is curved. An arc is part of the circle’s circumference."},{"text":"diameter","correct":false,"explanation":"A diameter is a straight line across the circle through its centre, not part of its curved edge. An arc is part of the circumference."},{"text":"area","correct":false,"explanation":"Area describes the two-dimensional space inside a circle, while an arc is a length along its boundary. It is therefore part of the circumference."}]},{"id":"d2657958-bee3-0912-5200-e36be1f8f79e","cardType":"basic","points":1,"front":"What fraction of a full circle is a sector with angle $\\theta$?","back":"$\\frac{\\theta}{360}$","mcqOptions":[{"text":"$\\frac{\\theta}{360}$","correct":true,"explanation":"A full circle contains $360^\\circ$, so a sector with angle $\\theta$ occupies the fraction $\\theta$ divided by $360$ of the complete circle."},{"text":"$\\frac{\\theta}{180}$","correct":false,"explanation":"Dividing by $180$ compares the sector with a semicircle rather than a full circle. Since a full turn is $360^\\circ$, the correct fraction is $\\frac{\\theta}{360}$."},{"text":"$\\frac{360}{\\theta}$","correct":false,"explanation":"This reverses the required fraction and can produce a value greater than $1$ for a minor sector. The sector’s fraction is $\\frac{\\theta}{360}$."},{"text":"$\\frac{\\theta}{2\\pi}$","correct":false,"explanation":"The denominator $2\\pi$ applies when the angle is measured in radians. Here $\\theta$ is compared with $360^\\circ$, so the fraction is $\\frac{\\theta}{360}$."}]},{"id":"37bf19cb-0b48-8921-c634-ff1decd9472b","cardType":"basic","points":1,"front":"What fraction of a full circle is a $120^\\circ$ sector?","back":"$\\frac{1}{3}$","mcqOptions":[{"text":"$\\frac{1}{3}$","correct":true,"explanation":"A full circle is $360^\\circ$, and $120 \\div 360 = \\frac{1}{3}$. Therefore, a $120^\\circ$ sector occupies one third of the circle."},{"text":"$\\frac{2}{3}$","correct":false,"explanation":"This is the fraction occupied by the remaining $240^\\circ$ major sector. The $120^\\circ$ sector itself is $\\frac{120}{360} = \\frac{1}{3}$."},{"text":"$\\frac{1}{6}$","correct":false,"explanation":"This results from incorrectly comparing $120^\\circ$ with $720^\\circ$ or halving the correct fraction. Dividing by $360^\\circ$ gives $\\frac{120}{360} = \\frac{1}{3}$."},{"text":"$\\frac{1}{4}$","correct":false,"explanation":"One quarter of a circle corresponds to $90^\\circ$, not $120^\\circ$. Since $120 \\div 360 = \\frac{1}{3}$, the sector is one third."}]},{"id":"8faca4fe-4d35-9ba5-dd3d-a8817dc05e09","cardType":"cloze","points":null,"front":"A sector is made from two [ ] and the [ ] between them.","back":"A sector is made from two radii and the arc between them.","mcqOptions":[{"text":"A sector is made from two radii and the arc between them.","correct":true,"explanation":"A sector is the region enclosed by two straight radii extending from the centre and the curved arc of the circumference joining their endpoints."},{"text":"A sector is made from two diameters and the arc between them.","correct":false,"explanation":"Diameters cross the entire circle and do not form the two sides of a general sector. A sector is bounded by two radii and the arc between them."},{"text":"A sector is made from two radii and the chord between them.","correct":false,"explanation":"A chord is a straight line joining two points on the circumference and helps bound a segment, not a sector. A sector uses the curved arc."},{"text":"A sector is made from two chords and the circumference between them.","correct":false,"explanation":"The straight boundaries of a sector must meet at the centre, so they are radii rather than chords. Its curved boundary is the arc between them."}]},{"id":"ab4d5b37-623b-0002-adba-324a4d5baed4","cardType":"cloze","points":null,"front":"A minor sector is [ ]; a major sector is [ ].","back":"A minor sector is less than half a circle; a major sector is more than half a circle.","mcqOptions":[{"text":"A minor sector is less than half a circle; a major sector is more than half a circle.","correct":true,"explanation":"A minor sector has a central angle below $180^\\circ$, so it occupies less than half the circle. A major sector exceeds $180^\\circ$ and occupies more than half."},{"text":"A minor sector is more than half a circle; a major sector is less than half a circle.","correct":false,"explanation":"This swaps the definitions of minor and major sectors. A minor sector is less than half a circle, while a major sector is more than half."},{"text":"A minor sector is exactly half a circle; a major sector is more than half a circle.","correct":false,"explanation":"Exactly half a circle is a semicircular sector with a $180^\\circ$ angle, not a minor sector. A minor sector must be less than half a circle."},{"text":"A minor sector is less than half a circle; a major sector is exactly half a circle.","correct":false,"explanation":"A sector occupying exactly half the circle has a $180^\\circ$ angle and is semicircular, not major. A major sector must occupy more than half the circle."}]},{"id":"58253c44-04e2-f87f-2d9e-7a11282b63b5","cardType":"basic","points":1,"front":"What formula finds the area of a sector with radius $r$ and angle $\\theta$?","back":"$\\frac{\\theta}{360}\\times \\pi r^2$","mcqOptions":[{"text":"$\\frac{\\theta}{360}\\times \\pi r^2$","correct":true,"explanation":"A sector occupies the fraction $\\frac{\\theta}{360}$ of a full circle, whose area is $\\pi r^2$, so these quantities are multiplied."},{"text":"$\\frac{\\theta}{180}\\times \\pi r^2$","correct":false,"explanation":"Using 180 as the denominator incorrectly treats a semicircle as the whole shape. A complete turn is $360^\\circ$, so the area is $\\frac{\\theta}{360}\\times \\pi r^2$."},{"text":"$\\frac{\\theta}{360}\\times 2\\pi r$","correct":false,"explanation":"The factor $2\\pi r$ is the circumference, so this formula calculates arc length rather than area. Sector area uses $\\frac{\\theta}{360}\\times \\pi r^2$."},{"text":"$\\frac{360}{\\theta}\\times \\pi r^2$","correct":false,"explanation":"The fraction has been inverted, making smaller angles produce larger areas. The sector’s fraction of the circle is $\\frac{\\theta}{360}$, giving $\\frac{\\theta}{360}\\times \\pi r^2$."}]},{"id":"c1e3cfd4-dac2-7316-b115-40f34ccde01d","cardType":"basic","points":2,"front":"What is the area of a sector with radius 8 cm and angle $135^\\circ$?","back":"$24\\pi\\text{ cm}^2$","mcqOptions":[{"text":"$24\\pi\\text{ cm}^2$","correct":true,"explanation":"The sector is $\\frac{135}{360}=\\frac{3}{8}$ of a circle. Multiplying $\\frac{3}{8}$ by $\\pi(8^2)=64\\pi$ gives $24\\pi\\text{ cm}^2$."},{"text":"$48\\pi\\text{ cm}^2$","correct":false,"explanation":"This result is twice the correct area, typically from using 180 instead of 360 as the full-turn denominator. Using $\\frac{135}{360}\\times\\pi(8^2)$ gives $24\\pi\\text{ cm}^2$."},{"text":"$19.2\\pi\\text{ cm}^2$","correct":false,"explanation":"This uses $135/360=3/8$ incorrectly as $3/10$. The correct calculation is $\\frac{3}{8}\\times64\\pi=24\\pi\\text{ cm}^2$."},{"text":"$32\\pi\\text{ cm}^2$","correct":false,"explanation":"This comes from treating $135^\\circ$ as half a circle. It is actually three-eighths of a full turn, so the area is $\\frac{3}{8}\\times64\\pi=24\\pi\\text{ cm}^2$."}]},{"id":"aab40550-71b8-3183-f6e7-c05a62eb98c7","cardType":"basic","points":1,"front":"What is $24\\pi$ to 3 significant figures?","back":"$75.4$","mcqOptions":[{"text":"$75.4$","correct":true,"explanation":"Calculating $24\\times\\pi$ gives approximately $75.3982$. The first three significant digits are 7, 5 and 3, and the following 9 rounds the result to $75.4$."},{"text":"$75.3$","correct":false,"explanation":"This truncates the decimal instead of rounding it. Since $24\\pi\\approx75.3982$, the fourth significant digit is 9, so the third digit rounds up to give $75.4$."},{"text":"$7.54$","correct":false,"explanation":"The decimal point has been shifted one place left, effectively dividing the value by 10. Since $24\\pi\\approx75.3982$, the correct three-significant-figure answer is $75.4$."},{"text":"$754$","correct":false,"explanation":"The decimal point has been shifted one place right, making the value ten times too large. As $24\\pi\\approx75.3982$, it rounds to $75.4$ to three significant figures."}]},{"id":"c44fb2cb-925a-3541-e6bf-7226a56223d2","cardType":"basic","points":1,"front":"If a sector angle is $220^\\circ$, what fraction should be used for its area?","back":"$\\frac{220}{360}$","mcqOptions":[{"text":"$\\frac{220}{360}$","correct":true,"explanation":"A full circle contains $360^\\circ$, so a $220^\\circ$ sector represents $220$ out of $360$ equal angular parts. Its required fraction is therefore $\\frac{220}{360}$."},{"text":"$\\frac{220}{180}$","correct":false,"explanation":"Using 180 as the denominator compares the sector with a semicircle rather than a full circle. Sector area must use the fraction $\\frac{220}{360}$."},{"text":"$\\frac{140}{360}$","correct":false,"explanation":"Subtracting $220^\\circ$ from $360^\\circ$ gives the complementary $140^\\circ$ sector, not the stated sector. The required area fraction is $\\frac{220}{360}$."},{"text":"$\\frac{360}{220}$","correct":false,"explanation":"The fraction has been inverted, producing a value greater than 1 even though the sector is smaller than a circle. The correct fraction is $\\frac{220}{360}$."}]},{"id":"65411a1f-d4de-582b-37ac-98543cfc27c2","cardType":"cloze","points":null,"front":"The arc length is the length of the [ ] of a sector.","back":"The arc length is the length of the curved edge of a sector.","mcqOptions":[{"text":"curved edge","correct":true,"explanation":"An arc is the curved part of a circle’s circumference between the two radii. Therefore, arc length measures the sector’s curved edge rather than either straight side."},{"text":"straight edge","correct":false,"explanation":"A sector has two straight edges, and each is a radius rather than an arc. Arc length specifically measures the curved edge forming part of the circumference."},{"text":"sector radius","correct":false,"explanation":"The radius measures the distance from the centre to the circumference, not distance around the circle. Arc length is the length of the sector’s curved edge."},{"text":"whole perimeter","correct":false,"explanation":"The perimeter also includes both straight radii, so it is not the same as arc length. Arc length measures only the curved edge of the sector."}]},{"id":"4cae073f-7641-b7e9-27a3-304d8f8753af","cardType":"basic","points":1,"front":"What formula finds arc length for radius $r$ and angle $\\theta$?","back":"$\\frac{\\theta}{360}\\times 2\\pi r$","mcqOptions":[{"text":"$\\frac{\\theta}{360}\\times 2\\pi r$","correct":true,"explanation":"The circumference of a full circle is $2\\pi r$. A sector with angle $\\theta$ contains the fraction $\\frac{\\theta}{360}$ of that circumference."},{"text":"$\\frac{\\theta}{360}\\times \\pi r^2$","correct":false,"explanation":"This is the formula for sector area because it uses the circle’s area, $\\pi r^2$. Arc length instead uses the circumference: $\\frac{\\theta}{360}\\times2\\pi r$."},{"text":"$\\frac{\\theta}{360}\\times \\pi r$","correct":false,"explanation":"The circumference is $2\\pi r$, not $\\pi r$, so omitting the factor of 2 halves every arc length. The correct formula is $\\frac{\\theta}{360}\\times2\\pi r$."},{"text":"$\\frac{360}{\\theta}\\times 2\\pi r$","correct":false,"explanation":"The angular fraction has been inverted, making shorter sectors produce larger lengths. A sector occupies $\\frac{\\theta}{360}$ of the circumference, giving $\\frac{\\theta}{360}\\times2\\pi r$."}]},{"id":"93a95dac-b7af-3e38-f71b-6a8ea71ca4d1","cardType":"basic","points":2,"front":"Find the arc length for radius 10 cm and angle $108^\\circ$.","back":"$6\\pi\\text{ cm}$","mcqOptions":[{"text":"$6\\pi\\text{ cm}$","correct":true,"explanation":"The sector is $\\frac{108}{360}=\\frac{3}{10}$ of a circle. Its arc is therefore $\\frac{3}{10}\\times2\\pi(10)=6\\pi\\text{ cm}$."},{"text":"$3\\pi\\text{ cm}$","correct":false,"explanation":"This result omits the factor of 2 from the circumference formula, using $\\pi r$ instead of $2\\pi r$. The correct arc length is $6\\pi\\text{ cm}$."},{"text":"$12\\pi\\text{ cm}$","correct":false,"explanation":"This doubles the circumference factor, effectively using $4\\pi r$ instead of $2\\pi r$. Applying $\\frac{108}{360}$ to the correct circumference gives $6\\pi\\text{ cm}$."},{"text":"$10\\pi\\text{ cm}$","correct":false,"explanation":"This treats $108^\\circ$ as half of a full turn. It is actually $\\frac{3}{10}$ of $360^\\circ$, so the arc length is $6\\pi\\text{ cm}$."}]},{"id":"a714338d-48ba-5a18-88d8-0a2cb6042a9d","cardType":"basic","points":1,"front":"If an answer is required “in terms of $\\pi$”, what should you do?","back":"Leave $\\pi$ in the answer.","mcqOptions":[{"text":"Leave $\\pi$ in the answer.","correct":true,"explanation":"An exact answer in terms of $\\pi$ retains the symbol rather than replacing it with a decimal approximation. Numerical factors may be simplified, but $\\pi$ remains present."},{"text":"Use $3.14$ in the answer.","correct":false,"explanation":"Replacing $\\pi$ with 3.14 gives an approximation rather than an exact answer in terms of $\\pi$. The symbol $\\pi$ should remain in the final expression."},{"text":"Round $\\pi$ in the answer.","correct":false,"explanation":"Rounding $\\pi$ produces an approximate value and removes the requested exact form. An answer in terms of $\\pi$ must leave $\\pi$ unchanged in the result."},{"text":"Replace $\\pi$ with a decimal.","correct":false,"explanation":"Converting to a decimal removes $\\pi$ and usually introduces rounding error. When the question requests an answer in terms of $\\pi$, retain the $\\pi$ symbol."}]},{"id":"f59c638e-b657-61d9-3b1f-410eb18bccff","cardType":"basic","points":1,"front":"What formula finds the perimeter of a sector from its radius and arc length?","back":"$\\text{perimeter}=\\text{arc length}+2r$","mcqOptions":[{"text":"$\\text{perimeter}=\\text{arc length}+2r$","correct":true,"explanation":"A sector’s boundary consists of its curved arc and two straight radii, so its perimeter is the arc length plus $2r$."},{"text":"$\\text{perimeter}=\\text{arc length}+r$","correct":false,"explanation":"This counts only one of the sector’s two straight radii. The complete boundary includes both radii, so the correct formula is $\\text{perimeter}=\\text{arc length}+2r$."},{"text":"$\\text{perimeter}=\\text{arc length}+2\\pi r$","correct":false,"explanation":"The term $2\\pi r$ is the circumference of a complete circle, not the sector’s straight edges. Those edges total $2r$, giving $\\text{perimeter}=\\text{arc length}+2r$."},{"text":"$\\text{perimeter}=\\text{arc length}+r^2$","correct":false,"explanation":"Squaring the radius produces an area quantity, so it cannot be added to an arc length. The two straight sides total $2r$, giving $\\text{perimeter}=\\text{arc length}+2r$."}]},{"id":"d3d388e1-cdef-8d00-6c66-7df5f35ca83d","cardType":"basic","points":2,"front":"A sector has radius 4.8 cm and arc length $2\\pi$ cm. What is its perimeter?","back":"$2\\pi+9.6\\text{ cm}$","mcqOptions":[{"text":"$2\\pi+9.6\\text{ cm}$","correct":true,"explanation":"The sector perimeter is its arc length plus two radii. Since $2r=2\\times4.8=9.6$ cm, the perimeter is $2\\pi+9.6\\text{ cm}$."},{"text":"$2\\pi+4.8\\text{ cm}$","correct":false,"explanation":"This adds only one radius to the arc length, missing the sector’s second straight side. Adding $2\\times4.8=9.6$ cm gives $2\\pi+9.6\\text{ cm}$."},{"text":"$4\\pi+9.6\\text{ cm}$","correct":false,"explanation":"This incorrectly doubles the given arc length from $2\\pi$ to $4\\pi$. Keeping the stated arc and adding both radii gives $2\\pi+9.6\\text{ cm}$."},{"text":"$2\\pi+19.2\\text{ cm}$","correct":false,"explanation":"This adds four radii rather than the sector’s two radii. The straight edges total only $2\\times4.8=9.6$ cm, so the perimeter is $2\\pi+9.6\\text{ cm}$."}]},{"id":"0fa462b7-d01e-bc12-d51c-b37834e1dc22","cardType":"cloze","points":null,"front":"For shaded regions, shaded area = [ ].","back":"For shaded regions, shaded area = total area - unshaded area.","mcqOptions":[{"text":"total area - unshaded area","correct":true,"explanation":"The total region is divided into shaded and unshaded parts. Subtracting the unshaded area from the total therefore leaves exactly the shaded area."},{"text":"total area + unshaded area","correct":false,"explanation":"Adding the unshaded area counts that region twice because it is already included in the total. The shaded area is found using total area - unshaded area."},{"text":"unshaded area - total area","correct":false,"explanation":"This reverses the subtraction and would normally produce a negative result. The larger total area must come first: total area - unshaded area."},{"text":"total area ÷ unshaded area","correct":false,"explanation":"Dividing two areas gives a ratio rather than the area of the shaded region. The required area is found using total area - unshaded area."}]},{"id":"2c67a73e-37ed-aaa8-158d-0c08138819df","cardType":"basic","points":1,"front":"If a semicircle has diameter 18 cm, what radius should be used?","back":"9 cm","mcqOptions":[{"text":"9 cm","correct":true,"explanation":"The radius is half the diameter. Dividing the 18 cm diameter by 2 gives the required radius of 9 cm."},{"text":"18 cm","correct":false,"explanation":"This uses the diameter directly instead of converting it to a radius. The radius is half of 18 cm, so the correct value is 9 cm."},{"text":"36 cm","correct":false,"explanation":"This doubles the diameter rather than halving it. Since radius equals diameter divided by 2, the radius to use is 9 cm."},{"text":"$9\\pi$ cm","correct":false,"explanation":"Multiplying by $\\pi$ is associated with circumference calculations, not converting diameter to radius. Halving the 18 cm diameter gives a radius of 9 cm."}]},{"id":"f999f581-f202-638a-3e64-f2f15e347a8a","cardType":"basic","points":1,"front":"What shortcut formula finds sector area when radius $r$ and arc length $L$ are given?","back":"$A=\\frac{1}{2}rL$","mcqOptions":[{"text":"$A=\\frac{1}{2}rL$","correct":true,"explanation":"A sector’s area can be found directly from its radius and arc length by taking half their product, so $A=\\frac{1}{2}rL$."},{"text":"$A=rL$","correct":false,"explanation":"This omits the factor of one-half and therefore gives twice the sector’s actual area. The correct shortcut is $A=\\frac{1}{2}rL$."},{"text":"$A=\\frac{1}{2}r^2L$","correct":false,"explanation":"This incorrectly squares the radius and produces dimensions of cubic length rather than area. Using radius and arc length requires $A=\\frac{1}{2}rL$."},{"text":"$A=2rL$","correct":false,"explanation":"This multiplies the radius-arc product by 2 instead of halving it, making the result four times too large. The correct formula is $A=\\frac{1}{2}rL$."}]},{"id":"e63bc2a6-2dad-9590-d64e-151dcf7f755f","cardType":"basic","points":2,"front":"Find the sector area when $r=8$ cm and $L=5\\pi$ cm.","back":"$20\\pi\\text{ cm}^2$","mcqOptions":[{"text":"$20\\pi\\text{ cm}^2$","correct":true,"explanation":"Using $A=\\frac{1}{2}rL$ gives $A=\\frac{1}{2}\\times8\\times5\\pi=20\\pi$. As this is an area, the units are $\\text{cm}^2$."},{"text":"$40\\pi\\text{ cm}^2$","correct":false,"explanation":"This calculates $rL$ but omits the factor of one-half, doubling the sector area. Halving $8\\times5\\pi=40\\pi$ gives $20\\pi\\text{ cm}^2$."},{"text":"$10\\pi\\text{ cm}^2$","correct":false,"explanation":"This effectively halves the product twice, making the result half the required area. Applying $A=\\frac{1}{2}rL$ once gives $20\\pi\\text{ cm}^2$."},{"text":"$160\\pi\\text{ cm}^2$","correct":false,"explanation":"This incorrectly squares the radius when using the radius-arc shortcut. The formula is $A=\\frac{1}{2}rL$, which gives $\\frac{1}{2}\\times8\\times5\\pi=20\\pi\\text{ cm}^2$."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"Sector Areas and Arc Lengths","title":"Sector Areas and Arc Lengths Flashcards","topbar":{"backHref":"/gcse/maths/ocr/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"cc48e296-482e-4466-b220-5fbf39ba2c14","walled":"$L39"}],"$L3a","$L3b"]