4-4-13-volume-of-a-prism/questions","formulaUrl":null,"initialIndex":24,"initialQuestionWindow":[{"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","block":{"nodes":[{"latex":"$$\\htmlData{anchor=pythagoras}{\\color{@sky}{x^2+6^2=10^2}}$"},{"latex":"$$\\htmlData{anchor=rearranged}{\\color{@sky}{x^2=10^2-6^2}}$"},{"latex":"$$\\htmlData{anchor=evaluated}{\\color{@amber}{x^2=100-36}}$"},{"latex":"$$\\htmlData{anchor=answer}{\\color{@green}{x=\\sqrt{64}=8\\text{ cm}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"pythagoras"},{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"rearranged","from":"pythagoras","color":"sky","label":"rearrange"},{"to":"evaluated","from":"rearranged","color":"amber","label":"evaluate squares"},{"to":"answer","from":"evaluated","color":"green","label":"square root"}]},"order":1,"title":"Apply Pythagoras","description":"Substitute the two known side lengths into Pythagoras' theorem, then solve for the base. Writing x^2 + 6^2 = 10^2 earns the Pythagoras M1. Reaching the correct base, x = 8 cm, earns A1."},{"type":"text","order":2,"title":"Calculate triangle area","content":"Now use the area formula for a triangle: $\\text{area}=\\frac{1}{2}\\times\\text{base}\\times\\text{perpendicular height}$. Substituting the lengths gives $\\frac{1}{2}\\times 8\\times 6=24\\text{ cm}^2$. This earns the cross-sectional area method mark.","highlights":[{"text":"cross-section in the shape of a right-angled triangle","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=formula}{\\color{@sky}{V=\\text{cross-sectional area}\\times\\text{length}}}$"},{"latex":"$$\\htmlData{anchor=substitution}{\\color{@amber}{V=24\\times12}}$"},{"latex":"$$\\htmlData{anchor=final}{\\color{@green}{V=288\\text{ cm}^3}}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"substitution"},{"color":"green","style":"text-color-with-background","anchor":"final"}],"connections":[{"to":"substitution","from":"formula"

Volume of a Prism

Question 19

Volume of a Prism Questions

Volume of a Prism

Question 19

Volume of a Prism Questions