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 tangent to a circle with centre O touches the circle at point P, where OP = 5 cm. Point Q lies on the tangent so that PQ = 12 cm. Using the fact that a tangent is perpendicular to the radius at the point of contact, work out the length OQ.
- 2.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.
- 3.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?
- 4.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.
- 5.A chord divides a circle into two segments of different sizes. Write down the name given to the smaller of the two segments.
- 6.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.
- 7.A sector of a circle has an angle of 90° at the centre. Write down what fraction of the whole circle this sector represents.
- 8.A diameter divides a circle into two equal halves. Write down the special name given to each of these two halves.
- 9.A chord is drawn across a circle, dividing the circle into two regions. Write down the name of one of these two regions.
- 10.A chord divides a circle into two regions, one larger than the other. Write down the term used for the smaller of these two regions.
- 11.Two points on a circle divide its circumference into two arcs of different lengths. Write down the name given to the shorter of the two arcs.
- 12.A straight line is drawn from the centre of a circle to a point on the circle. Write down the name of this line.
- 13.A circle has centre O. Radii OA and OB are drawn, together with the arc AB, enclosing a sector. The straight chord AB is also drawn. Write down the name of the region enclosed between the chord AB and the arc AB.
- 14.A tangent touches a circle with centre O at the point P. Q is a point on the tangent. Write down the circle fact that tells you the size of angle OPQ.
- 15.A circle has diameter 10 cm. Using π = 3.14, work out the circumference of the circle.
Answer key
- (d) 13 — Method: OP and PQ meet at a right angle because of the tangent–radius fact, so triangle OPQ is right-angled at P; use Pythagoras' theorem. Working: OQ² = OP² + PQ² = 5² + 12² = 25 + 144 = 169; OQ = √169 = 13. A student who answers 17 has simply added the two given lengths (5 + 12) instead of using Pythagoras' theorem. A student who answers 7 has subtracted the two given lengths (12 − 5) instead of using Pythagoras' theorem. A student who answers 144 has correctly squared 12 but stopped there, forgetting to add 5² and take the square root. Answer: 13 cm.
- (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.
- (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.
- (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) 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) 1/4 — A full turn at the centre of a circle is 360°, so a sector's fraction of the circle is its angle divided by 360°: 90 ÷ 360 = 1/4. 1/2 would be the fraction for a sector with an angle of 180°, not 90°. 3/4 is the fraction of the rest of the circle, the major sector left over from the 270° that is not part of this sector. 1/8 would be the fraction for a sector with an angle of 45°, half of 90°.
- (b) Semicircle — Method: recall the special name for exactly half of a circle. Working: a diameter splits the circle into two identical halves, and each half has its own specific name. A student who answers segment has confused this special case with the general region cut off by any chord. A student who answers sector has confused it with the pie-slice shape made by two radii. A student who answers arc has only named the curved edge, not the whole two-dimensional region. Answer: semicircle.
- (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.
- (c) minor segment — A chord splits a circle into two segments; the smaller of the two is called the minor segment and the larger one the major segment. "major segment" names the larger region, the opposite of what is asked for. "sector" is a different region altogether, enclosed by two radii and an arc, not by a chord. "arc" is a curved length along the circumference, not an enclosed region at all.
- (d) Minor arc — Method: compare the lengths of the two arcs formed by the two points, and recall the term for the shorter one. Working: the two points split the circumference into two arcs; the shorter one is the minor arc and the longer one is the major arc. A student who answers major arc has picked the longer arc by mistake. A student who answers minor segment has confused the curved boundary with the enclosed two-dimensional region. A student who answers chord has named the straight line joining the two points instead of the curved arc. Answer: minor arc.
- (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) segment — The region enclosed between a chord and the arc it cuts off is called a segment. A sector is the different region already named in the question, enclosed by the two radii and the arc, not by the chord. A tangent is a straight line that touches the circle at one point, not an enclosed region at all. The circumference is the total distance around the whole circle, a length, not a region.
- (d) The angle between a tangent and a radius is 90° — OP is the radius drawn to the point of contact P, and a circle theorem states that a tangent always meets that radius at a right angle, so angle OPQ = 90°. A tangent is not parallel to the radius it touches — at the point of contact it is perpendicular to that radius, not parallel to it. A tangent does not pass through the centre — a straight line through the centre that also touches the circle at one point would have to be a diameter, which is a different line entirely. The angle-in-a-semicircle theorem needs a triangle drawn inside the circle with a diameter as its longest side; there is no such triangle here, just a tangent and a radius.
- (b) 31.4 cm — Circumference = π × diameter, so 3.14 × 10 = 31.4 cm. 15.7 cm comes from using the radius, 5 cm, in place of the diameter: 3.14 × 5 = 15.7, which is only half the circumference. 78.5 cm comes from using the area formula π × radius² instead of the circumference formula: 3.14 × 5² = 3.14 × 25 = 78.5. 62.8 cm comes from keeping the 2 from the radius form of the formula, C = 2 × π × radius, but putting the full diameter into it: 2 × 3.14 × 10 = 62.8.
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