Are angles in polygons on the Foundation paper as well as Higher?

Yes. G3 is a crossover reference, so both tiers are assessed on the interior angle sum and on exterior angles adding to 360 degrees. What separates a Higher version is the number of steps: two regular polygons joined along a shared side, an unknown number of sides recovered from a single angle, or interior angles written as expressions in x that have to be solved.

Is the formula for the sum of the interior angles given in the Edexcel exam?

No. The Exam Aid sheet issued with every 1MA1 paper carries the quadratic formula, Pythagoras, the trigonometric ratios and rules, the compound interest formula, the two probability rules and the trapezium, prism and circle formulae, but nothing at all about polygon angles. You need 180(n - 2) for the interior angle sum, and 360 divided by n for the exterior angle of a regular polygon, from memory.

How do you find the number of sides from an interior angle?

Take the interior angle away from 180 degrees to get the exterior angle, then divide 360 by that. An interior angle of 162 degrees gives an exterior angle of 18 degrees, and 360 divided by 18 is 20 sides. Dividing 360 by the interior angle instead is the standard slip and returns about 2.2, which should tell you straight away that something is wrong.

How many marks is an angles in polygons question worth on Edexcel Higher?

A single step, such as the exterior angle of a regular polygon, is 1-2 marks and tends to sit early in the paper. Questions that combine two polygons, or that hide the missing angle in an irregular one, run to 3-5 marks: there are method marks for the angle sum and for a correct subtraction before the accuracy mark for the angle itself.

Angles in Polygons - Edexcel GCSE Higher Maths

Typically 2-5 marks per paper, usually one multi-step question · Spec 1MA1

The interior angles of a polygon with $n$ sides add to $180(n - 2)$ degrees, and however irregular the shape, its exterior angles add to $360^\circ$ . Those two facts sit at spec reference G3 on Edexcel's 1MA1, a crossover point assessed at both tiers, so a Higher paper spends no time on the definitions. It chains them instead: sides to an angle, an angle back to a number of sides, and a regular polygon to the isosceles triangles hiding inside it once a diagonal is drawn. Neither formula appears on the Exam Aid sheet issued with the paper, so both are recall.

Edexcel sets G3 on all three papers. Paper 1 is non-calculator and the numbers there divide cleanly, which is a hint in itself: if $360$ refuses to divide by the exterior angle you have found, check the angle before you blame the division. Papers 2 and 3 can afford a 14-sided or 22-sided polygon whose interior angle needs rounding to 1 decimal place. Two question shapes recur. One gives an irregular polygon with every angle listed but one and asks for the missing angle; the other joins regular shapes edge to edge, an octagon against a pentagon or three tiles meeting at a point, and asks for the angle in the gap. Polygon diagrams come with the usual warning that they are not accurately drawn, so a hexagon that looks regular is only regular when the wording says so.

Marks go to a written route rather than a remembered number. Putting $(6 - 2) \times 180 = 720$ on the page before you touch the given angles secures the first mark even if the subtraction afterwards goes wrong, whereas a total recalled straight from memory and mistyped as 620 scores nothing at all. The two losses that repeat are dividing by the number of sides in an irregular polygon, where nothing entitles you to equal angles, and dividing the leftover by the number of sides when only two of the angles were unknown. Drill the reverse route until it runs without thinking: interior angle, subtract from $180^\circ$ , divide the result into $360^\circ$ .

Worked example

A square, a regular hexagon and a regular polygon P are placed flat so that one vertex of each meets at the point O. There are no gaps and no overlaps between the three shapes at O. Work out the number of sides of polygon P.

[4 marks]

Angles at a point add to $360^\circ$ , so the three interior angles meeting at O total $360^\circ$ . The square contributes $90^\circ$ .
B1: angles at a point quoted, with the square's interior angle written as 90
For the regular hexagon the exterior angle is $360 \div 6 = 60^\circ$ , so its interior angle is $180 - 60 = 120^\circ$ .
M1: interior angle of the hexagon obtained from its exterior angle
The interior angle of P is $360 - 90 - 120 = 150^\circ$ , so the exterior angle of P is $180 - 150 = 30^\circ$ .
M1: subtracting to leave P's interior angle, then converting it to the exterior angle
The exterior angles of P add to $360^\circ$ , so P has $360 \div 30 = 12$ sides.
A1: 12; dividing 360 by the interior angle instead would give 2.4, which is not a polygon

Practice questions

These are original questions in Edexcel 1MA1 Higher style. Nothing here is drawn, so every figure is spelled out: which polygons are regular, how many sides they have, and which side or vertex two of them share. Most answers are whole numbers of degrees so you can work without a calculator, and the handful that are not state the rounding your answer has to match.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

Question 1 Warm-up [2 marks]

A regular polygon has $n$ sides. Write an expression, in terms of $n$ , for the size in degrees of one exterior angle.

Your expression (equivalent forms accepted, e.g. 2(x+3) or 2x+6)
Question 2 Exam pace [2 marks]

The interior angles of a polygon add up to $1440^\circ$ . Work out the number of sides of the polygon.

Your answer (fractions like 1/2 are fine)
Question 3 Exam pace [3 marks]

The base of a gazebo is a regular polygon with 14 sides. Work out the size of one interior angle of the base, in degrees, correct to 1 decimal place.

Your answer (fractions like 1/2 are fine)
Question 4 Stretch [4 marks]

Two identical regular pentagons and one regular polygon P are placed flat so that one vertex of each meets at the same point, with no gaps and no overlaps. Work out the number of sides of polygon P.

Your answer (fractions like 1/2 are fine)
Question 5 Stretch [4 marks]

Polygon A and polygon B are both regular polygons. Polygon B has 6 sides. Each interior angle of polygon A is $30^\circ$ larger than each interior angle of polygon B. Work out the number of sides of polygon A.

Your answer (fractions like 1/2 are fine)
Question 6 Stretch [4 marks]

The sum of the interior angles of a regular polygon is 8 times the sum of its exterior angles. Work out the size of one interior angle of the polygon, in degrees.

Your answer (fractions like 1/2 are fine)

Common mistakes examiners see

Using $n \times 180^\circ$ for the interior angle sum, so a hexagon comes out as $1080^\circ$ rather than $720^\circ$ and every angle built on it is wrong.

Write $180(n - 2)$ and say why the 2 is there: diagonals from one vertex cut the polygon into $n - 2$ triangles. Test it on a shape you already know, since a quadrilateral must give $2 \times 180 = 360^\circ$ .
Dividing $360^\circ$ by the number of sides and labelling the result the interior angle. For a regular decagon that produces $36^\circ$ , which is the exterior angle.

$360 \div n$ is always the exterior angle. Take it away from $180^\circ$ for the interior angle, then check the size is believable: past four sides, every interior angle of a regular polygon is more than $90^\circ$ .
Dividing the interior angle sum by the number of sides when the polygon is irregular. A pentagon with angles of $95^\circ$ , $140^\circ$ , $102^\circ$ and $86^\circ$ given does not have a fifth angle of $540 \div 5 = 108^\circ$ .

Divide only when the question uses the word regular. Otherwise subtract the listed angles from the sum: $540 - 423 = 117^\circ$ .
Subtracting from the sum correctly, then dividing the remainder by the number of sides instead of by the number of unknown angles. A pentagon with three angles given and two equal unknowns leaves a remainder shared between two angles, not five.

Count the unknowns before you divide and write the division out in words first. Then substitute your angle back and add all five: they have to reach $540^\circ$ .
Accepting a number of sides that is not a whole number. An interior angle of $155^\circ$ gives an exterior angle of $25^\circ$ and $360 \div 25 = 14.4$ , which gets rounded to 14 and written on the answer line.

Sides come in whole numbers. If the division is not exact, re-check the interior-to-exterior step; if that step is sound, the honest answer is that no regular polygon has that angle.

Frequently asked questions

Are angles in polygons on the Foundation paper as well as Higher?: Yes. G3 is a crossover reference, so both tiers are assessed on the interior angle sum and on exterior angles adding to 360 degrees. What separates a Higher version is the number of steps: two regular polygons joined along a shared side, an unknown number of sides recovered from a single angle, or interior angles written as expressions in $x$ that have to be solved.
Is the formula for the sum of the interior angles given in the Edexcel exam?: No. The Exam Aid sheet issued with every 1MA1 paper carries the quadratic formula, Pythagoras, the trigonometric ratios and rules, the compound interest formula, the two probability rules and the trapezium, prism and circle formulae, but nothing at all about polygon angles. You need $180(n - 2)$ for the interior angle sum, and 360 divided by n for the exterior angle of a regular polygon, from memory.
How do you find the number of sides from an interior angle?: Take the interior angle away from 180 degrees to get the exterior angle, then divide 360 by that. An interior angle of 162 degrees gives an exterior angle of 18 degrees, and 360 divided by 18 is 20 sides. Dividing 360 by the interior angle instead is the standard slip and returns about 2.2, which should tell you straight away that something is wrong.
How many marks is an angles in polygons question worth on Edexcel Higher?: A single step, such as the exterior angle of a regular polygon, is 1-2 marks and tends to sit early in the paper. Questions that combine two polygons, or that hide the missing angle in an irregular one, run to 3-5 marks: there are method marks for the angle sum and for a correct subtraction before the accuracy mark for the angle itself.

Angles in Polygons - Edexcel GCSE Higher Maths

Worked example

Practice questions

Common mistakes examiners see

Frequently asked questions

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