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Only the surfaces exposed on the exterior are included."},{"text":"inside angles","correct":false,"explanation":"Angles describe turns between edges and do not have an area. The correct term is outside faces, because surface area totals the exterior surfaces of the shape."},{"text":"vertical edges","correct":false,"explanation":"Edges are one-dimensional lengths, so adding them would not give an area. Surface area is the total area of all the outside faces."},{"text":"corner points","correct":false,"explanation":"Corner points, or vertices, have no area and are counted rather than measured. The required term is outside faces, whose areas combine to give surface area."}]},{"id":"d3faaa19-efd1-0cea-9d5e-6f79c228b2f8","cardType":"basic","points":1,"front":"Surface area is measured in what type of units?","back":"Square units, e.g. cm² or m².","mcqOptions":[{"text":"Square units, e.g. cm² or m².","correct":true,"explanation":"Area measures two-dimensional space, so each length unit is multiplied by another length unit. This produces square units such as cm² or m²."},{"text":"Linear units, e.g. cm or m.","correct":false,"explanation":"Linear units measure one-dimensional quantities such as length or perimeter. Surface area is two-dimensional, so it is measured in square units such as cm² or m²."},{"text":"Cubic units, e.g. cm³ or m³.","correct":false,"explanation":"Cubic units measure three-dimensional volume, not the area covering a shape. Surface area is measured in square units such as cm² or m²."},{"text":"Angle units, e.g. degrees or radians.","correct":false,"explanation":"Degrees and radians measure angles rather than the size of surfaces. Surface area measures two-dimensional regions, so square units such as cm² or m² are required."}]},{"id":"e0f65344-56c6-6f13-1d4a-c8ff3b165762","cardType":"basic","points":1,"front":"What formula is used for the area of a rectangular face?","back":"$\\text{length}\\times\\text{width}$","mcqOptions":[{"text":"$\\text{length}\\times\\text{width}$","correct":true,"explanation":"A rectangle’s area is found by multiplying its two perpendicular dimensions. Therefore, the area of a rectangular face is length multiplied by width."},{"text":"$2\\times\\text{length}+2\\times\\text{width}$","correct":false,"explanation":"This adds the four side lengths and therefore calculates the rectangle’s perimeter. Its area is found using length multiplied by width."},{"text":"$\\frac{1}{2}\\times\\text{length}\\times\\text{width}$","correct":false,"explanation":"The factor of one-half belongs to the triangle area formula. A rectangular face occupies twice that triangular area, so its formula is length multiplied by width."},{"text":"$\\text{length}+\\text{width}$","correct":false,"explanation":"Adding the dimensions does not calculate either the complete perimeter or the enclosed area. The area of a rectangular face is length multiplied by width."}]},{"id":"5ade66b1-f48b-974d-46b9-849286ff212a","cardType":"cloze","points":null,"front":"The perpendicular height is measured [ ], not along a slanted side.","back":"The perpendicular height is measured at 90° to the base, not along a slanted side.","mcqOptions":[{"text":"at 90° to the base","correct":true,"explanation":"Perpendicular lines meet at a right angle, so the perpendicular height must be measured at 90° to the base rather than along a sloping side."},{"text":"parallel to the base","correct":false,"explanation":"A line parallel to the base never forms the required height between the base and opposite point. The perpendicular height is measured at 90° to the base."},{"text":"along the longest side","correct":false,"explanation":"The longest side may be slanted and therefore may not represent the shortest distance from the base. The perpendicular height is measured at 90° to the base."},{"text":"at 45° to the base","correct":false,"explanation":"A 45° line is diagonal rather than perpendicular, so using it gives an incorrect height. The perpendicular height must be measured at 90° to the base."}]},{"id":"faaffd00-0e3c-92ac-e193-70b569af17f1","cardType":"cloze","points":null,"front":"Triangle area: [ ], where $h$ is the perpendicular height.","back":"Triangle area: $A=\\frac{1}{2}bh$, where $h$ is the perpendicular height.","mcqOptions":[{"text":"$$A=\\frac{1}{2}bh$","correct":true,"explanation":"A triangle occupies half the area of a rectangle with the same base and perpendicular height. Its area is therefore one-half multiplied by base multiplied by height."},{"text":"$$A=bh$","correct":false,"explanation":"This calculates the area of a rectangle or parallelogram with base $b$ and height $h$. A triangle has half that area, so $A=\\frac{1}{2}bh$."},{"text":"$$A=\\frac{1}{2}(b+h)$","correct":false,"explanation":"Adding base and height combines lengths and cannot produce square units. Triangle area requires multiplying them and halving the result: $A=\\frac{1}{2}bh$."},{"text":"$$A=2bh$","correct":false,"explanation":"This doubles the corresponding rectangle’s area instead of taking half of it. A triangle with base $b$ and perpendicular height $h$ has area $A=\\frac{1}{2}bh$."}]},{"id":"68a65de7-3e97-45cd-ecfe-89d2430a6760","cardType":"cloze","points":null,"front":"Trapezium area: [ ], where $a$ and $b$ are parallel sides.","back":"Trapezium area: $A=\\frac{1}{2}(a+b)h$, where $a$ and $b$ are parallel sides.","mcqOptions":[{"text":"$$A=\\frac{1}{2}(a+b)h$","correct":true,"explanation":"The area of a trapezium equals its average parallel-side length, $\\frac{a+b}{2}$, multiplied by its perpendicular height. This gives $A=\\frac{1}{2}(a+b)h$."},{"text":"$$A=(a+b)h$","correct":false,"explanation":"This uses the sum rather than the average of the parallel sides, making the area twice as large. The correct formula is $A=\\frac{1}{2}(a+b)h$."},{"text":"$$A=\\frac{1}{2}abh$","correct":false,"explanation":"Multiplying both parallel sides together introduces an extra length factor and gives cubic units. Their lengths must be added, so $A=\\frac{1}{2}(a+b)h$."},{"text":"$$A=\\frac{1}{2}(a-b)h$","correct":false,"explanation":"Subtracting the parallel sides finds their difference, not their average length. They must be added before multiplying by half the height: $A=\\frac{1}{2}(a+b)h$."}]},{"id":"b5a89d65-45aa-2a45-7b1c-ff1bbcf84223","cardType":"basic","points":2,"front":"A trapezium has parallel sides 12 cm and 6 cm, height 5 cm. What is its area?","back":"45 cm²","mcqOptions":[{"text":"45 cm²","correct":true,"explanation":"Using $A=\\frac{1}{2}(a+b)h$ gives $\\frac{1}{2}(12+6)\\times5=\\frac{1}{2}\\times18\\times5=45$ cm²."},{"text":"90 cm²","correct":false,"explanation":"This omits the factor of one-half and calculates $(12+6)\\times5$. Averaging the parallel sides gives $\\frac{1}{2}(12+6)\\times5=45$ cm²."},{"text":"30 cm²","correct":false,"explanation":"This uses only the 12 cm side and halves $12\\times5$, ignoring the other parallel side. The trapezium formula gives $45$ cm²."},{"text":"15 cm²","correct":false,"explanation":"This incorrectly uses half the difference between the parallel sides: $\\frac{1}{2}(12-6)\\times5$. Their lengths must be added, giving $45$ cm²."}]},{"id":"894c566e-6330-3ebd-dea6-2a3b2fee87d4","cardType":"basic","points":2,"front":"Why do you double the three different face areas for a cuboid?","back":"Opposite faces are the same size.","mcqOptions":[{"text":"Opposite faces are the same size.","correct":true,"explanation":"A cuboid has three pairs of opposite faces. The two faces in each pair have identical dimensions and areas, so each different face area is counted twice."},{"text":"Adjacent faces are the same shape.","correct":false,"explanation":"Adjacent faces can have different dimensions and areas, even though each is rectangular. Doubling works because opposite faces are the same size and form three matching pairs."},{"text":"Opposite edges are the same length.","correct":false,"explanation":"Equal opposite edge lengths help define the cuboid, but they do not directly explain doubling face areas. Each area is doubled because opposite faces are the same size."},{"text":"All six faces have the same area.","correct":false,"explanation":"All six faces have equal area only for a cube, not for every cuboid. A general cuboid has three pairs in which opposite faces are the same size."}]},{"id":"fd2b8489-f834-2560-45a9-2af4e5ae2834","cardType":"cloze","points":null,"front":"Cuboid total surface area: [ ].","back":"Cuboid total surface area: $2lw+2lh+2wh$.","mcqOptions":[{"text":"$$2lw+2lh+2wh$","correct":true,"explanation":"A cuboid has two faces of each area $lw$, $lh$ and $wh$, so its total surface area is $2lw+2lh+2wh$."},{"text":"$$lw+lh+wh$","correct":false,"explanation":"This counts only one face of each size. A cuboid has two of every face, so the correct formula is $2lw+2lh+2wh$."},{"text":"$$2lw+lh+wh$","correct":false,"explanation":"This doubles only the $lw$ face, leaving the other two pairs undercounted. All three face areas must be doubled, giving $2lw+2lh+2wh$."},{"text":"$$2l+2w+2h$","correct":false,"explanation":"This adds edge lengths rather than face areas, so it cannot calculate surface area. Multiplying pairs of dimensions gives the correct formula $2lw+2lh+2wh$."}]},{"id":"57252c23-1b73-dab4-164e-de9b3d53f101","cardType":"basic","points":2,"front":"A cuboid is 9 cm by 4 cm by 3 cm. What is its total surface area?","back":"150 cm²","mcqOptions":[{"text":"150 cm²","correct":true,"explanation":"The three different face areas are $9\\times4=36$, $9\\times3=27$ and $4\\times3=12$. Doubling their sum gives $2(36+27+12)=150$ cm²."},{"text":"75 cm²","correct":false,"explanation":"This adds one face of each size but forgets that every face has an opposite matching face. Doubling $36+27+12$ gives the correct total of 150 cm²."},{"text":"108 cm²","correct":false,"explanation":"This is the volume, found by multiplying all three dimensions, rather than the total area of the faces. The surface area is $2(36+27+12)=150$ cm²."},{"text":"138 cm²","correct":false,"explanation":"This counts only one $4\\times3$ face instead of two. Including both of every face gives $72+54+24=150$ cm²."}]},{"id":"f4dac5a2-bda6-d544-998b-91ec176e569a","cardType":"cloze","points":null,"front":"A prism has the same shape all the way through; the repeated end shape is the [ ].","back":"A prism has the same shape all the way through; the repeated end shape is the cross-section.","mcqOptions":[{"text":"cross-section","correct":true,"explanation":"The cross-section is the repeated two-dimensional end shape that remains identical along the entire length of a prism."},{"text":"side elevation","correct":false,"explanation":"A side elevation is a two-dimensional view of an object, not necessarily its repeated end shape. The repeated shape running through a prism is its cross-section."},{"text":"surface area","correct":false,"explanation":"Surface area is the total area of all the prism’s outer faces, not the repeated end shape. That repeated shape is called the cross-section."},{"text":"longitudinal section","correct":false,"explanation":"A longitudinal section is a cut made along an object’s length. The identical end shape repeated through a prism is instead called its cross-section."}]},{"id":"89f505b3-6a29-405e-9668-b5f03f14d48c","cardType":"basic","points":1,"front":"What formula gives the total surface area of any prism?","back":"$2\\times\\text{cross-section area}+\\text{perimeter}\\times\\text{length}$","mcqOptions":[{"text":"$2\\times\\text{cross-section area}+\\text{perimeter}\\times\\text{length}$","correct":true,"explanation":"The two congruent ends contribute twice the cross-section area, while unfolding the side faces produces rectangles with combined area equal to perimeter times length."},{"text":"$\\text{cross-section area}+\\text{perimeter}\\times\\text{length}$","correct":false,"explanation":"This includes only one of the two congruent end faces. Both ends must be counted, so the formula begins with $2\\times\\text{cross-section area}$."},{"text":"$2\\times\\text{cross-section area}+\\text{perimeter}+\\text{length}$","correct":false,"explanation":"Adding perimeter and length does not produce an area because both are one-dimensional measurements. Their product gives the area of the rectangular side faces."},{"text":"$2\\times\\text{cross-section area}+2\\times\\text{perimeter}\\times\\text{length}$","correct":false,"explanation":"This doubles the side-face area unnecessarily. The perimeter already includes every edge around the cross-section, so the correct side-face term is just perimeter times length."}]},{"id":"cc766973-3cc3-50cb-7914-b2f7ff40b3d3","cardType":"basic","points":1,"front":"In the prism formula, what does $\\text{perimeter}\\times\\text{length}$ find?","back":"The rectangular side faces.","mcqOptions":[{"text":"The rectangular side faces.","correct":true,"explanation":"Each cross-section edge forms a rectangle whose other dimension is the prism’s length. Adding these rectangles gives the cross-section perimeter multiplied by the length."},{"text":"The two cross-section faces.","correct":false,"explanation":"The two end faces are found using twice the cross-section area. Perimeter multiplied by length instead gives the combined area of the rectangular side faces."},{"text":"The volume inside the prism.","correct":false,"explanation":"Volume is found by multiplying cross-section area by length. Multiplying the cross-section perimeter by length finds the combined area of the rectangular side faces."},{"text":"The perimeter of each end.","correct":false,"explanation":"The perimeter is already one factor in the calculation. Multiplying it by the prism’s length converts the boundary strips into the total area of the rectangular side faces."}]},{"id":"3823c475-add5-d802-d5fb-cc8bbd29611f","cardType":"basic","points":2,"front":"Triangular prism: cross-section perimeter 36 cm, length 7 cm. What is the side-face area?","back":"252 cm²","mcqOptions":[{"text":"252 cm²","correct":true,"explanation":"The side-face area of a prism is cross-section perimeter multiplied by length. Therefore, $36\\times7=252$, giving a side-face area of 252 cm²."},{"text":"43 cm²","correct":false,"explanation":"This comes from adding 36 and 7 instead of multiplying them. Side-face area is perimeter times length, so $36\\times7=252$ cm²."},{"text":"126 cm²","correct":false,"explanation":"This incorrectly halves the product, perhaps treating the side faces like a triangle. They are rectangles, so the full calculation is $36\\times7=252$ cm²."},{"text":"504 cm²","correct":false,"explanation":"This doubles the product unnecessarily. The perimeter already includes every side of the triangular cross-section, so $36\\times7=252$ cm² is the complete side-face area."}]},{"id":"74d2daff-2c0d-12a4-cef7-b16eb3c9d653","cardType":"basic","points":2,"front":"How do you find the area of an L-shaped end made by removing a corner from a rectangle?","back":"Subtract the missing rectangle’s area.","mcqOptions":[{"text":"Subtract the missing rectangle’s area.","correct":true,"explanation":"The L-shape is the complete outer rectangle with a smaller rectangular corner removed, so subtracting the missing rectangle’s area leaves the required L-shaped area."},{"text":"Add the missing rectangle’s area.","correct":false,"explanation":"Adding would include the removed corner and make the result larger than the outer rectangle. The missing rectangle’s area must be subtracted from the outer rectangle’s area."},{"text":"Subtract the missing rectangle’s perimeter.","correct":false,"explanation":"Perimeter measures boundary length and cannot be subtracted from an area. Find both rectangular areas, then subtract the missing rectangle’s area from the outer rectangle’s area."},{"text":"Multiply the missing rectangle’s area.","correct":false,"explanation":"Multiplying areas does not represent removing a corner and produces the wrong units. Subtract the missing rectangle’s area from the area of the complete outer rectangle."}]},{"id":"5fd0ce54-f047-eb6c-6b4d-4241fef6e033","cardType":"basic","points":2,"front":"An L-shaped end fits inside 8 cm by 7 cm, with a 3 cm by 4 cm corner removed. What is its area?","back":"44 cm²","mcqOptions":[{"text":"44 cm²","correct":true,"explanation":"The complete outer rectangle has area $8\\times7=56$ cm², and the removed corner has area $3\\times4=12$ cm². Therefore, $56-12=44$ cm²."},{"text":"68 cm²","correct":false,"explanation":"This adds the missing corner’s area to the outer rectangle instead of removing it. The correct calculation is $56-12=44$ cm²."},{"text":"12 cm²","correct":false,"explanation":"This is only the area of the removed $3$ cm by $4$ cm corner. Subtracting it from the outer area gives $56-12=44$ cm²."},{"text":"56 cm²","correct":false,"explanation":"This is the area of the complete $8$ cm by $7$ cm rectangle and ignores the removed corner. Subtracting $12$ cm² gives 44 cm²."}]},{"id":"12de5365-b8e6-dc4f-f2cc-396096264ed9","cardType":"basic","points":2,"front":"Why trace around a composite cross-section when finding perimeter?","back":"To include the step edges.","mcqOptions":[{"text":"To include the step edges.","correct":true,"explanation":"Tracing the outside boundary ensures that every exposed edge is counted, including the short step edges that are easy to overlook in a composite cross-section."},{"text":"To include the internal joins.","correct":false,"explanation":"Internal joins are not exposed parts of the perimeter and must not be counted. Tracing is used to include all outside edges, especially the step edges."},{"text":"To include the face areas.","correct":false,"explanation":"Face areas are needed for surface area, not perimeter. Tracing the boundary helps identify all exposed lengths, including the step edges."},{"text":"To include the diagonal lengths.","correct":false,"explanation":"Diagonal lengths are not automatically part of a composite cross-section’s boundary. The purpose of tracing is to ensure the step edges are included."}]},{"id":"add8af3f-db72-820d-aa8f-7e93f6bcf3d0","cardType":"basic","points":1,"front":"How many circular faces does a closed cylinder have?","back":"Two equal circular faces.","mcqOptions":[{"text":"Two equal circular faces.","correct":true,"explanation":"A closed cylinder has a circular face at each end. These two circles have the same radius, so they are equal in area."},{"text":"One circular face.","correct":false,"explanation":"One circular face describes an open cylinder with one end missing. A closed cylinder has two equal circular faces, one at each end."},{"text":"Three equal circular faces.","correct":false,"explanation":"The curved surface is not a circular face. A closed cylinder has exactly two equal circular faces, located at its two ends."},{"text":"Two unequal circular faces.","correct":false,"explanation":"A cylinder’s two end circles must be equal because they share the same radius. A closed cylinder therefore has two equal circular faces."}]},{"id":"a289480e-023a-00af-199c-eb7d9f292a88","cardType":"cloze","points":null,"front":"Closed cylinder total surface area: [ ].","back":"Closed cylinder total surface area: $2\\pi r^2+2\\pi rh$.","mcqOptions":[{"text":"$$2\\pi r^2+2\\pi rh$","correct":true,"explanation":"The two circular ends have combined area $2\\pi r^2$, while the curved surface has area $2\\pi rh$. Adding them gives the total surface area."},{"text":"$$\\pi r^2+2\\pi rh$","correct":false,"explanation":"This includes only one circular end, omitting the second area of $\\pi r^2$. A closed cylinder requires $2\\pi r^2+2\\pi rh$."},{"text":"$$2\\pi r^2$","correct":false,"explanation":"This adds the two circular ends but omits the curved surface. The curved area is $2\\pi rh$, so the full formula is $2\\pi r^2+2\\pi rh$."},{"text":"$$2\\pi r^2+\\pi rh$","correct":false,"explanation":"The curved surface uses the circumference $2\\pi r$ multiplied by height, not $\\pi r$ multiplied by height. The correct total is $2\\pi r^2+2\\pi rh$."}]},{"id":"bff68b29-c359-7cb7-3b8a-60f1ae3c3ab2","cardType":"basic","points":1,"front":"If a cylinder question gives the diameter, what must you do first?","back":"Halve it to get the radius.","mcqOptions":[{"text":"Halve it to get the radius.","correct":true,"explanation":"The radius is the distance from the centre to the circumference, which is half the diameter. Therefore, divide the given diameter by two first."},{"text":"Double it to get the radius.","correct":false,"explanation":"Doubling confuses the relationship between radius and diameter. The diameter is already twice the radius, so it must be halved to obtain the radius."},{"text":"Square it to get the radius.","correct":false,"explanation":"Squaring the diameter does not convert it into a radius; squaring is used later in circle-area calculations. First halve the diameter to get the radius."},{"text":"Use it as the radius.","correct":false,"explanation":"Using the diameter unchanged as the radius doubles the required length and makes area calculations four times too large. The diameter must be halved."}]},{"id":"485afd41-e2fe-7839-8550-11c216d29cc2","cardType":"basic","points":1,"front":"If area is in cm², how do you convert it to m²?","back":"Divide by 10000.","mcqOptions":[{"text":"Divide by 10000.","correct":true,"explanation":"Since $1\\text{ m}=100\\text{ cm}$, squaring the conversion gives $1\\text{ m}^2=10000\\text{ cm}^2$. Therefore, divide a value in cm² by 10000."},{"text":"Divide by 100.","correct":false,"explanation":"Dividing by 100 applies the linear length conversion without squaring it. Area conversion requires dividing by $100^2=10000$ to change cm² into m²."},{"text":"Multiply by 10000.","correct":false,"explanation":"Multiplying by 10000 converts from m² to cm², the opposite direction. To convert the smaller cm² units into m², divide by 10000."},{"text":"Divide by 1000000.","correct":false,"explanation":"Dividing by 1000000 incorrectly uses the cubic conversion associated with cm³ and m³. For square units, divide by $100^2=10000$."}]},{"id":"e64615f2-a17d-da6a-0079-709643c92431","cardType":"basic","points":1,"front":"In a paint question, what do you do after dividing by coverage per pot?","back":"Round up to the next whole pot.","mcqOptions":[{"text":"Round up to the next whole pot.","correct":true,"explanation":"Any positive fraction of an additional pot means another complete pot must be bought. Rounding up ensures there is enough paint for the entire area."},{"text":"Round down to the previous whole pot.","correct":false,"explanation":"Rounding down would leave too little paint whenever the result is not already a whole number. The number of pots must always be rounded up."},{"text":"Round normally to the nearest whole pot.","correct":false,"explanation":"Rounding to the nearest whole number can round down and leave insufficient paint. Since only complete pots can be bought, always round up."},{"text":"Leave it as a decimal number of pots.","correct":false,"explanation":"A decimal result represents the calculated proportion of a pot, but shops sell whole pots. It must be rounded up to the next whole pot."}]},{"id":"7c36da12-777e-742e-6c60-885a0c98b35c","cardType":"basic","points":2,"front":"Four boxes need 10.24 m² of paint. One pot covers 3 m². How many pots are needed?","back":"4 pots","mcqOptions":[{"text":"4 pots","correct":true,"explanation":"Dividing the required area by the coverage gives $10.24\\div3\\approx3.41$. Since only whole pots can be bought, this must be rounded up to 4 pots."},{"text":"3 pots","correct":false,"explanation":"Three pots cover only $3\\times3=9$ m², which is less than 10.24 m². The calculation gives about 3.41, so 4 pots are required."},{"text":"3.41 pots","correct":false,"explanation":"This is the unrounded result of $10.24\\div3$, but paint is purchased in complete pots. About 3.41 pots must be rounded up to 4 pots."},{"text":"5 pots","correct":false,"explanation":"Rounding $10.24\\div3\\approx3.41$ up gives 4, not 5. Buying five pots would provide more complete pots than the calculation requires."}]}]}],"examCountdown":null,"heading":"Surface Area","title":"Surface Area Flashcards","topbar":{"backHref":"/gcse/maths/ccea/flashcards"},"topicSidebar":"$L39"}]}],"topicId":"35c42827-ae12-f47c-7aae-496a1878027d","walled":"$L3a"}],"$L3b","$L3c"]