Circle vocabulary and perimeter

(a) Segment

1. A sector has two straight radius boundaries.
2. The boundary here is one chord and one arc, so the region is a segment.

• B1 Correct answer: Segment

(a) Tangent

1. A tangent touches the circle once and does not cross it.

• B1 Correct answer: Tangent.

(b) Segment

1. A chord cuts the circle into two segments.

• B1 Correct answer: Segment.

(c) Diameter

1. The longest chord, through the centre, is a diameter.

• B1 Correct answer: Diameter.

(a) 30.8 cm

1. The curved edge is half the circumference.
   $$\tfrac{1}{2} \times \pi \times 12 = 6\pi = 18.849\ldots$$
2. Add the straight edge, the diameter.
   $$6\pi + 12 = 30.849\ldots$$
3. So the perimeter is 30.8 cm (3 significant figures).

• P1 Half the circumference, $6\pi$ or $18.8\ldots$
• P1 Adding the diameter to the arc.
• A1 30.8 cm (awrt 30.8).

(a) $14\pi$

1. $$14\pi$$
2. The circumference of a circle is C = $\pi d.$
3. Substitute d = 14: C = $14\pi$ m.

• P1 Establishing $14\pi$ or an equivalent valid method.
• A1 Correct answer: $14\pi$

(a) $4\pi +8$

1. Half the circumference is the curved border.
   $$2\pi \times 4/2$$
2. Add the diameter to close the boundary.
   $$4\pi +8$$
3. Therefore $4\pi +8$.

• P1 Half the circumference is the curved border.
• P1 Add the diameter to close the boundary.
• A1 Correct answer: $4\pi +8$

(b) A chord joins two points on the circumference. A diameter does this and also passes through the centre.

1. A chord joins two points on the circumference. A diameter does this and also passes through the centre.

• C1 Correct conclusion with supporting reasoning: A chord joins two points on the circumference. A diameter does this and also passes through the centre.

(a) $250\pi$

1. Calculate the distance for one turn using the diameter.
   $$\pi \times 10$$
2. Multiply by the number of full turns.
   $$25\times 10\pi$$
3. Therefore $250\pi$.

• P1 Calculate the distance for one turn using the diameter.
• P1 Multiply by the number of full turns.
• A1 Correct answer: $250\pi$

(b) $\frac{5}{2}\pi$

1. Divide centimetres by 100.
   $$\frac{250\pi}{100}$$
2. Therefore $\frac{5}{2}\pi$.

• P1 Divide centimetres by 100.
• A1 Correct answer: $\frac{5}{2}\pi$

(a) $6\pi$

1. Subtract circumference A from circumference B.
   $$16\pi -2\pi \times 5$$
2. Therefore $6\pi$.

• M1 Subtract circumference A from circumference B.
• A1 Correct answer: $6\pi$

(b) $16$ cm

1. Therefore $16$ cm.

• B1 Correct answer: $16$ cm

(a) $r = 38.2$ m

1. The straights give 160 m, so the curved ends give
   $$400 - 160 = 240$$
2. Two semicircles make a full circle.
   $$2\pi r = 240$$
3. Divide.
   $$r = \frac{240}{2\pi} = 38.197\ldots$$

• P1 Finding the curved length, 240 m.
• P1 Forming $2\pi r = 240$ (or $\pi d = 240$).
• A1 38.2 m (awrt 38.2).

(b) 7.54 m

1. Only the curved parts change. The full circle's circumference drops from $2\pi r$ to $2\pi(r - 1.2)$.
2. The difference does not depend on $r$.
   $$2\pi r - 2\pi(r - 1.2) = 2\pi \times 1.2 = 7.539\ldots$$

• M1 $2\pi \times 1.2$, or subtracting two full lap lengths correctly.
• A1 7.54 m (awrt 7.54).