Printable · GCSE Foundation · ages 14-16
Circle definitions and parts of a circle worksheet — GCSE Foundation
Fifteen questions on "circle definitions and parts of a circle" — DfE statement G9. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
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Circle definitions and parts of a circle worksheet — GCSE Foundation
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- 1.A trundle wheel has a diameter of 0.5 m. It is rolled along the ground and makes 20 complete turns. Using π = 3.14, work out the total distance rolled, in metres.
- 2.A line segment joins two points on the circumference of a circle, without necessarily passing through the centre. Write down the general term for this line segment.
- 3.In one circle, chord PQ is 6 cm long, chord RS is 10 cm long, chord TU is 14 cm long and chord VW is 15 cm long. Write down which of these chords lies closest to the centre of the circle.
- 4.Two circles are drawn with the same centre but different radii. Write down the term used to describe this pair of circles.
- 5.A chord is drawn across a circle, dividing the circle into two regions. Write down the name of one of these two regions.
- 6.A straight line is drawn from the centre of a circle to a point on the circle. Write down the name of this line.
- 7.Write down the term for the distance all the way around the edge of a circle.
- 8.A circular table mat has diameter 20 cm. Yasmin has a 70 cm length of ribbon and sews it around the edge of the mat with no overlap. Using π = 3.14, work out how much ribbon is left over after going around the mat once.
- 9.A sector of a circle is bounded by two straight edges and one curved edge. Write down the mathematical term for each of the two straight edges.
- 10.A straight line touches the edge of a circle at exactly one point and does not cross into the circle at all. Write down the term for this line.
- 11.A circle has two straight radii drawn from the centre to the edge, together with the arc between them. Write down the name of the region enclosed by these two radii and the arc.
- 12.A chord divides a circle into two segments of different sizes. Write down the name given to the smaller of the two segments.
- 13.The curved part of a circle's circumference, between two points on the circle, has its own name. Write down this name.
- 14.A cyclist rides around a perfectly circular track with centre O. At every moment of the ride, the distance from O to the cyclist is exactly the same. Which term names this fixed distance?
- 15.A circle has a diameter of 4 cm. Work out the radius of the circle.
Answer key
- (a) 31.4 — Method: first find the circumference of one turn using π × diameter, then multiply by the number of turns. Working: circumference = 3.14 × 0.5 = 1.57 m; total distance = 1.57 × 20 = 31.4 m. A student who answers 15.7 has mistakenly halved the diameter again before multiplying, using 0.25 m instead of 0.5 m. A student who answers 3.14 has simply written down π itself, without completing the circumference or multiplying by the number of turns. A student who answers 62.8 has mistakenly doubled the diameter to 1 m before multiplying, treating the given length as a radius. Answer: 31.4 m.
- (a) Chord — Method: identify what the two endpoints of the line segment are, and whether it must pass through the centre. Working: a line joining any two points on the circumference, whether or not it passes through the centre, is a chord, and the diameter is just a special case of it. A student who answers diameter has wrongly assumed the line must pass through the centre. A student who answers radius has confused joining two circumference points with joining the centre to the circumference. A student who answers tangent has confused a line crossing through the circle with one that only touches its outside. Answer: chord.
- (c) VW — Method: within one circle a chord's distance from the centre is fixed by its length, because a chord passing nearer the centre cuts further across the circle; so the chord that lies closest to the centre is simply the longest one listed. Working: the four lengths are 6 cm, 10 cm, 14 cm and 15 cm. Placing them in order, the greatest is 15 cm, and that length belongs to VW, so VW lies closest to the centre. Answer: VW. The distractors: PQ comes from reversing the rule and taking the shortest chord to be the one tucked nearest the centre; TU comes from knowing that the very longest chord is a diameter, deciding that such a chord passes through the centre rather than lying close to it, ruling the 15 cm chord out on that ground and taking the next longest; RS comes from reading 'closest to the centre' as 'nearest the middle of the list of lengths' and picking a middling value.
- (d) Concentric circles — Method: focus on what the two circles have in common — their centre, not their size. Working: both circles share exactly the same centre point but have different radii, which is the defining feature of this pair of circles. A student who answers congruent circles has confused 'same centre' with 'same size', but congruent circles simply have equal radii and need not share a centre. A student who answers tangential circles has confused circles that touch each other at one point with ones that share a centre. A student who answers similar circles has used the general term for the same shape at different sizes, missing the specific 'same centre' fact. Answer: concentric circles.
- (a) segment — A chord cuts a circle into two regions, and each of these regions is called a segment. A sector is a different region, bounded by two radii and an arc, not by a chord. An arc is a curved part of the circumference, a length, not a region. A radius is a straight line from the centre to the edge, also a length, not a region.
- (b) The radius — Method: each of the four names describes a line in a fixed position relative to the circle, so match the description in the question against those positions. Working: the line described has one end at the centre and its other end on the circle. A chord joins two points that both lie on the circle, so it is ruled out. A diameter joins two points on the circle by passing through the centre, so it is twice as long as the line described. A tangent touches the circle at exactly one point and never reaches the centre. The line with one end at the centre and one end on the circle is a radius. Answer: The radius. The distractors: The diameter comes from remembering only that the line involves the centre, and not noticing that it stops there instead of carrying on to the far side; The chord comes from seeing one end on the circle and assuming both ends are; The tangent comes from matching the phrase 'a point on the circle' to the line that touches at exactly one point, ignoring that a tangent never reaches the centre.
- (b) circumference — The distance all the way around the outside edge of a circle is called the circumference. The diameter is a straight line across the circle through the centre, so it is a length through the circle, not around it. The radius is a straight line from the centre to the edge, again a length across, not around. The area is the amount of surface inside the circle, a region, not a length at all.
- (a) 7.2 cm — The distance around the mat is the circumference, π × diameter = 3.14 × 20 = 62.8 cm. Yasmin started with 70 cm, so the ribbon left over is 70 − 62.8 = 7.2 cm. 62.8 cm is the circumference itself, the amount of ribbon used, not what is left over. 50 cm comes from subtracting the diameter instead of the circumference: 70 − 20 = 50. 20 cm is simply the diameter of the mat and involves no calculation with the ribbon length at all.
- (c) Radius — Method: recall that a sector is formed using two straight lines drawn from the centre out to the circle's edge. Working: each straight edge of a sector runs from the centre of the circle to a point on the circumference. A student who answers chord has confused a straight edge from the centre with one joining two points on the circumference. A student who answers diameter has wrongly assumed the two straight edges must form a single full diameter. A student who answers arc has named the curved edge instead of the straight edges. Answer: radius (radii).
- (d) tangent — A line that touches a circle at exactly one point, without crossing into the circle, is called a tangent. A chord is a straight line joining two points ON the circle, so it touches at two points, not one. A radius runs from the centre to the circle's edge, not along the outside of it. A diameter is a chord that passes through the centre, also touching the circle at two points.
- (d) sector — The region enclosed by two radii and the arc between them is called a sector. A segment is a different region, enclosed by a CHORD and an arc — it does not touch the centre at all. A chord is a straight line, not a region, so it cannot be the answer to a question asking for a region. A semicircle is only correct when the angle between the two radii is exactly 180°, which is not stated here, so it is too specific to be the general answer.
- (b) Minor segment — Method: compare the sizes of the two regions cut off by the chord, and recall the term used for the smaller one. Working: the chord creates two segments; the smaller region is called the minor segment and the larger one the major segment. A student who answers major segment has picked the larger region by mistake instead of the smaller one. A student who answers minor arc has named the curved boundary rather than the two-dimensional region it encloses. A student who answers semicircle has wrongly assumed the chord must pass through the centre. Answer: minor segment.
- (a) arc — A curved part of a circle's circumference, between two points, is called an arc. A chord is the straight line joining those two points, not the curved part. A segment is the region enclosed between a chord and an arc, and a sector is the region enclosed between two radii and an arc — both segment and sector are regions, not curved lengths.
- (a) radius — The distance from the centre O to any point on the circle is the radius. The diameter is the distance right across the circle, through the centre, from one side to the other — twice the radius, not the same measurement. The circumference is the total distance around the circle, not from the centre. A chord is a line joining two points on the circle that does not have to pass through the centre at all.
- (b) 2 cm — Method: a diameter is made of two radii end to end, so going back from a diameter to a radius undoes that doubling, which gives radius = diameter ÷ 2. Working: the diameter is 4 cm, so the radius is 4 ÷ 2 = 2 cm. Answer: 2 cm. The distractors: 8 cm comes from multiplying by 2 instead of dividing by it, the relationship applied in the wrong direction; 1 cm comes from halving twice, once to reach the radius and then once more as though a second halving were called for; 0.5 cm comes from writing the division upside down as 2 ÷ 4 rather than 4 ÷ 2.
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