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Because it is **regular**, all $6$ exterior angles are equal.","highlights":[{"text":"regular hexagon","location":"specification"}]},{"type":"text","order":1,"title":"Use the angle sum","content":"The exterior angles of any polygon add up to $360^\\circ$. Since the hexagon is regular, divide this total equally between its $6$ exterior angles."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\dfrac{\\htmlData{anchor=turn}{\\color{@sky}{360^\\circ}}}{\\htmlData{anchor=sides}{\\color{@sky}{6}}}$"},{"latex":"$$\\htmlData{anchor=answer}{\\color{@green}{60^\\circ}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"turn"},{"color":"sky","style":"text-color","anchor":"sides"},{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"answer","from":"turn","color":"green","label":"divide by 6"}]},"order":2,"title":"Divide by six","description":"A regular hexagon has six equal exterior angles, so divide the full turn by six. The calculation earns M1, and the displayed final answer of 60 degrees earns A1."}]}],"generatedAt":"2026-07-20T11:07:23.629Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"4174438a-15b0-45d7-9048-a2b5588228f2:exterior_angle_sum","label":"Using Exterior Angle Sum","steps":[{"type":"text","order":0,"title":"Identify the number of sides","content":"A hexagon has **6** sides. Because it is a **regular** hexagon, all of its sides and angles are equal. This means all of its exterior angles are also equal to one another."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Sum of exterior angles} = \\htmlData{anchor=sum}{\\color{@amber}{360^\\circ}}$"},{"latex":"$$\\text{Number of sides } (n) = \\htmlData{anchor=sides}{\\color{@sky}{6}}$"},{"latex":"$$\\text{Exterior angle} = \\frac{\\htmlData{anchor=sum-sub}{\\color{@amber}{360^\\circ}}}{\\htmlData{anchor=sides-sub}{\\color{@sky}{6}}}$"},{"latex":"$$\\phantom{\\text{Exterior angle}} = \\htmlData{anchor=ans}{\\color{@purple}{60^\\circ}}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"sum"},{"color":"sky","style":"text-color","anchor":"sides"},{"color":"purple","style":"box","anchor":"ans"}],"connections":[{"to":"sum-sub","from":"sum","color":"amber","label":null,"style":"solid"},{"to":"sides-sub","from":"sides","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Calculate the exterior angle","description":"The sum of the exterior angles of any polygon is always $360^\\circ$. We divide this sum by the number of sides to find the size of one exterior angle."},{"type":"text","order":2,"title":"Understand the marks","content":"In your exam, you earn marks as follows:\n- **1 method mark (M1)** for showing the division: $$\\frac{360}{6}$$\n- **1 accuracy mark (A1)** for writing down the correct final answer: $$60^\\circ$$"}]},{"id":"4174438a-15b0-45d7-9048-a2b5588228f2:interior_angle_subtraction","label":"Via Interior Angle Calculation","steps":[{"type":"text","order":0,"title":"Identify the number of sides","content":"A hexagon is a polygon with $6$ sides. Since it is a **regular** hexagon, all of its $6$ sides are equal in length, and all of its $6$ interior angles are equal in size.\n\nWe will denote the number of sides as:\n$$n = 6$$"},{"type":"text","order":1,"title":"Find total interior sum","content":"To find the size of a single interior angle, we first calculate the sum of all interior angles in an $n$-sided polygon using the formula:\n$$\\text{Sum of interior angles} = (n - 2) \\times 180^\\circ$$\n\nSubstituting $n = 6$ into the formula:\n$$\\begin{aligned} \\text{Sum} &= (6 - 2) \\times 180^\\circ \\\\ &= 4 \\times 180^\\circ \\\\ &= 720^\\circ \\end{aligned}$$"},{"type":"text","order":2,"title":"Calculate single interior angle","content":"Since a regular hexagon has $6$ equal interior angles, we find the size of one interior angle by dividing the total sum ($720^\\circ$) by $6$:\n\n$$\\text{Interior angle} = \\frac{720^\\circ}{6} = 120^\\circ$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"An interior angle and its adjacent exterior angle lie on a straight line, which means they must add up to $180^\\circ$."},{"latex":"$$\\text{Exterior angle} + \\htmlData{anchor=int-angle}{\\color{@sky}{\\text{Interior angle}}} = 180^\\circ$"},{"latex":"$$\\text{Exterior angle} + \\htmlData{anchor=int-val}{\\color{@sky}{120^\\circ}} = 180^\\circ$"},{"latex":"$$\\text{Exterior angle} = 180^\\circ - 120^\\circ = 60^\\circ$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"int-angle"},{"color":"sky","style":"text-color","anchor":"int-val"}],"connections":[{"to":"int-val","from":"int-angle","color":"sky","label":"substitute","style":"solid"}]},"order":3,"title":"Calculate the exterior angle","description":"Subtract the interior angle from $180^\\circ$ to find the exterior angle."},{"type":"text","order":4,"title":"State the final answer","content":"The size of an exterior angle of a regular hexagon is **$60^\\circ$**."}]}],"generatedAt":"2026-06-12T10:54:39.916Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":57},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/review/68d99764-ab3b-4751-8ae1-86e44dd6c9ea.mp4","durationSeconds":67,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/review/68d99764-ab3b-4751-8ae1-86e44dd6c9ea.jpg"}},{"id":"8871d123-e7f3-4bfa-b326-61318547e0de","revisionId":"3fce1b18-802a-4c17-a9e1-f9d4c4590957","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"easy","tier":null,"totalMarks":2,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"5dbffe16-1910-40c7-af79-3c5a15b93391","label":"a","content":"The size of each exterior angle in a regular polygon is $30^\\circ$.\nWork out how many sides the polygon has.","markscheme":"- $\\frac{360}{30}$ M1\n- $12$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Evidence of dividing 360 by the exterior angle.","code":"M","acceptable":["360 / 30"]},{"id":"p2","marks":1,"description":"Correct number of sides.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-06T14:00:33.486Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"divide_exterior_sum","label":"Divide the exterior-angle sum","steps":[{"type":"text","order":0,"title":"Recall the angle sum","content":"The exterior angles of any polygon add up to $360^\\circ$. Because the polygon is **regular**, all its exterior angles have the same size.","highlights":[{"text":"regular polygon","location":"specification"},{"text":"$$30^\\circ$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$n=\\htmlData{anchor=quotient}{\\dfrac{\\color{@sky}{360^\\circ}}{\\color{@amber}{30^\\circ}}}$"},{"latex":"$$\\htmlData{anchor=result}{n=12}$"}],"highlights":[{"color":"teal","style":"text-color-with-background","anchor":"quotient"},{"color":"green","style":"text-color-with-background","anchor":"result"}],"connections":[{"to":"result","from":"quotient","color":"teal","label":"calculate"}]},"order":1,"title":"Divide by one angle","description":"The exterior angles of any polygon add to 360°, so divide this total by the size of one exterior angle. There is one exterior angle per side. Writing the division earns M1. The value n = 12 earns A1, so the polygon has 12 sides."}]}],"generatedAt":"2026-07-19T18:20:42.374Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"5dbffe16-1910-40c7-af79-3c5a15b93391:exterior_angle_formula","label":"Using Exterior Angles","steps":[{"type":"text","order":0,"title":"Identify exterior angle sum","content":"A key property of any polygon is that the sum of its exterior angles is always $360^\\circ$.\n\nFor a **regular polygon**, all exterior angles are equal in size."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\text{Number of sides } (n) = \\frac{\\htmlData{anchor=sum}{\\color{@sky}{360^\\circ}}}{\\htmlData{anchor=ext}{\\color{@amber}{\\text{Size of each exterior angle}}}}$$ "}],"highlights":[{"color":"sky","style":"underline","anchor":"sum"},{"color":"amber","style":"underline","anchor":"ext"}],"connections":[]},"order":1,"title":"Recall the formula","description":"We can find the number of sides $n$ by dividing the total sum of the exterior angles ($360^\\circ$) by the size of one exterior angle."},{"type":"text","order":2,"title":"Calculate the sides","content":"Given that each exterior angle is $30^\\circ$, we substitute this into our formula:\n\n$$\\text{Number of sides} = \\frac{360^\\circ}{30^\\circ}$$\n\n$$\\text{Number of sides} = 12$$\n\nThis division earns the first mark (Method mark, **M1**), and the correct final answer of $12$ earns the second mark (Accuracy mark, **A1**).","calculator":[{"gdc":{"expression":"360 / 30","instructions":"Type `360 / 30` and press ENTER."},"ti84":{"expression":"360 / 30","instructions":"Type `360 / 30` and press ENTER."},"desmos":{"expression":"360 / 30","instructions":"Enter `360 / 30` in the input bar."},"description":"Perform the division to find the number of sides."}],"highlights":[{"text":"30^\\circ","location":"specification"}]}]},{"id":"5dbffe16-1910-40c7-af79-3c5a15b93391:interior_angle_formula","label":"Using Interior Angles","steps":[{"type":"text","order":0,"title":"Calculate the interior angle","content":"Since angles on a straight line add up to $180^\\circ$, an interior angle and an exterior angle of any polygon must add up to $180^\\circ$.\n\nWe are given that each exterior angle is $30^\\circ$. Let's calculate the size of each interior angle:\n\n$$\\text{Interior angle} = 180^\\circ - 30^\\circ = 150^\\circ$$","highlights":[{"text":"each exterior angle in a regular polygon is 30°","location":"specification"}]},{"type":"text","order":1,"title":"Set up the equation","content":"The formula for the size of each interior angle of a regular polygon with $n$ sides is:\n\n$$\\text{Interior angle} = \\frac{(n - 2) \\times 180^\\circ}{n}$$\n\nWe know the interior angle is $150^\\circ$. Substituting this into our formula gives:\n\n$$\\frac{(n - 2) \\times 180}{n} = 150$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=eq1}{\\frac{(n - 2) \\times 180}{n}} = \\htmlData{anchor=val1}{150}$"},{"latex":"$$(n - 2) \\times 180 = 150n$"},{"latex":"$$180n - 360 = 150n$"},{"latex":"$$\\htmlData{anchor=eq2}{30n} = \\htmlData{anchor=val2}{360}$"},{"latex":"$$n = 12$"}],"highlights":[{"color":"sky","style":"box","anchor":"eq1"},{"color":"sky","style":"box","anchor":"eq2"}],"connections":[{"to":"eq2","from":"eq1","color":"sky","label":"rearrange","style":"solid"}]},"order":2,"title":"Solve for n","description":"Rearrange the equation to find the number of sides $n$."},{"type":"text","order":3,"title":"State the final answer","content":"The regular polygon has **12** sides (which is a dodecagon)."}]}],"generatedAt":"2026-06-12T22:23:00.646Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":79},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/review/1c77a96b-3fe7-426b-b95d-0fb4f3a61090.mp4","durationSeconds":58,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/review/1c77a96b-3fe7-426b-b95d-0fb4f3a61090.jpg"}},{"id":"98db4511-c4a1-4577-aa85-0afb9a582380","revisionId":"6f1a918b-a75d-4751-8a0e-9249ed214581","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"easy","tier":null,"totalMarks":2,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"f74f189a-6709-4f03-8500-886096789c4f","label":null,"content":"The size of each exterior angle in a regular polygon is $24^\\circ$.\nWork out how many sides the polygon has.","markscheme":"- $\\frac{360}{24}$ M1\n- $15$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Divides $360$ by $24$.","code":"M","acceptable":["$$\\frac{360}{24}$","$$360 \\div 24$"]},{"id":"p2","marks":1,"description":"Gives the correct number of 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Angles in Polygons

Question 2

Angles in Polygons Questions

Angles in Polygons

Question 2

Angles in Polygons Questions