of the cube be $s$, the area of one face is $s^2$.\n\nTherefore, the formula for the total surface area ($A$) of a cube is:\n\n$$A = 6s^2$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$6\text{s}^2 = \\htmlData{anchor=area-val}{\\color{@sky}{294}}$$"},{"latex":"$$$\\text{s}^2 = \\frac{\\color{@sky}{294}}{6}$$"},{"latex":"$$$\\text{s}^2 = \\htmlData{anchor=face-area}{\\color{@amber}{49}}$$"},{"latex":"$$$\\text{s} = \\sqrt{\\color{@amber}{49}} = \\htmlData{anchor=side-len}{\\color{@purple}{7}} \\text{ cm}$$"}],"highlights":[{"color":"sky","style":"box","anchor":"area-val"},{"color":"amber","style":"box","anchor":"face-area"},{"color":"purple","style":"box","anchor":"side-len"}],"connections":[{"to":"face-area","from":"area-val","color":"sky","label":"divide by 6","style":"solid"},{"to":"side-len","from":"face-area","color":"amber","label":"square root","style":"solid"}]},"order":1,"title":"Solve for side length","description":"We set our surface area equation equal to the given surface area of $294\\text{ cm}^2$ to find the side length $s$."},{"type":"text","order":2,"title":"Calculate the volume","content":"The formula for the volume ($V$) of a cube is given by:\n\n$$V = s^3$$\n\nSubstituting our side length $s = 7\\text{ cm}$ into this formula:\n\n$$\\begin{aligned} V &= 7^3 \\\\ V &= 7 \\times 7 \\times 7 \\\\ V &= 343 \\text{ cm}^3 \\end{aligned}$$"},{"type":"text","order":3,"title":"Markscheme breakdown","content":"Here is how the **$4$ marks** are distributed for this question:\n* **M1**: Setting up the division to find the area of one face ($294 \\div 6$ or $s^2 = 49$)\n* **M1**: Finding the square root to calculate the side length ($s = 7\\text{ cm}$)\n* **M1**: Setting up the calculation for the volume ($7 \\times 7 \\times 7$)\n* **A1**: Reaching the correct final answer of $343\\text{ cm}^3$ with correct units"}]}],"generatedAt":"2026-06-12T23:07:33.996Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"cd1d4c6d-951d-43fc-af28-69810f56ed1e","revisionId":"a0b2889e-b761-4411-83d0-0e8b55eb02f3","questionNumber":null,"questionType":"LA","specification":"A glass prism has a cross-section in the shape of a right-angled triangle. The hypotenuse of the triangle is $10 \\text{ cm}$ and its height is $6 \\text{ cm}$. The length of the prism is $12 \\text{ cm}$.","difficulty":3,"difficultyBand":"hard","totalMarks":5,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"3ee79456-09df-4125-b1ec-a8f1e9c75962","label":"a","content":"Calculate the volume of the prism.","markscheme":"- Uses Pythagoras to find the missing side (base): $x^2 + 6^2 = 10^2$ M1\n- $x = \\sqrt{100 - 36} = 8$ A1\n- Area of cross-section = $\\frac{6 \\times 8}{2} = 24 \\text{ cm}^2$ M1\n- Volume = $24 \\times 12$ M1\n- Final answer = $288 \\text{ cm}^3$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Correct Pythagoras setup: $10^2 - 6^2$.","code":"M"},{"id":"p2","marks":1,"description":"Finds the missing triangle side length as 8.","code":"A"},{"id":"p3","marks":1,"description":"Calculation of the area of the triangular face.","code":"M"},{"id":"p4","marks":1,"description":"Calculation of the volume using Area × length.","code":"M"},{"id":"p5","marks":1,"description":"Final numerical value for volume.","code":"A"}],"finalAnswers":["288 \\text{ cm}^3","288"],"generatedAt":"2026-05-06T14:05:40.142Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"pythagoras_then_volume","label":"Pythagoras then volume","steps":[{"type":"text","order":0,"title":"Identify the triangle","content":"The prism has a triangular cross-section. The hypotenuse is $10\\text{ cm}$ and the perpendicular height is $6\\text{ cm}$, so use Pythagoras to find the missing base before finding the triangle's area.","highlights":[{"text":"The hypotenuse of the triangle is $10 \\text{ cm}$ and its height is $6 \\text{ cm}$.","location":"specification"}]},{"type":"hyper_latex"

Volume of a Prism

Question 5

Volume of a Prism Questions

Volume of a Prism

Question 5

Volume of a Prism Questions