Circles, Arcs & Sectors Flashcards

All 19 cards in this deck

What is the radius of a circle?

The distance from the centre to any point on the circumference.

What is the diameter of a circle?

A straight line from one side of the circle to the other, passing through the centre. It is twice the radius.

What is a chord of a circle?

A straight line joining two points on the circumference (it does not have to pass through the centre).

What is a tangent to a circle?

A straight line that touches the circle at exactly one point.

What is an arc of a circle?

A part of the circumference (a curved section of the edge).

True or false? A sector is the region between a chord and an arc.

False. That is a segment; a sector is the region between two radii and an arc.

What is the formula for the circumference of a circle?

$C = 2\pi r = \pi d$

What is the formula for the area of a circle?

$A = \pi r^2$

True or false? For a circle of diameter $10$ cm, the area is $\pi \times 10^2$.

False. You must halve the diameter first: area $= \pi \times 5^2$.

What are the steps to find the radius of a circle from its circumference?

e.g. circumference $= 12\pi$ cm

1. Divide the circumference by $\pi$ to get the diameter: $12\pi \div \pi = 12$ cm

2. Halve the diameter: $r = 6$ cm

What are the steps to find the perimeter of a semicircle?

e.g. radius $4$ cm

1. Half the circumference: $\frac{1}{2} \times 2\pi \times 4 = 4\pi$ cm

2. Add the diameter (the straight edge): $4\pi + 8$ cm

What are the steps to work out how many boxes of seed are needed to cover a circular lawn?

e.g. lawn of radius $10$ m, one box covers $50$ m$^2$

1. Area of circle: $\pi \times 10^2 = 314.1...$ m$^2$

2. Divide by the coverage: $314.1... \div 50 = 6.28...$

3. Round up to a whole box: $7$ boxes

What is the formula for the arc length of a sector of angle $\theta$?

e.g. angle $90^\circ$, radius $6$ cm

Arc length $= \frac{\theta}{360} \times 2\pi r$

$\frac{90}{360} \times 2\pi \times 6 = 3\pi$ cm

What is the formula for the area of a sector of angle $\theta$?

e.g. angle $90^\circ$, radius $6$ cm

Area $= \frac{\theta}{360} \times \pi r^2$

$\frac{90}{360} \times \pi \times 6^2 = 9\pi$ cm$^2$

What lengths make up the perimeter of a sector?

e.g. sector of radius $5$ cm with arc length $7$ cm

The arc length plus the two radii.

$7 + 5 + 5 = 17$ cm

What are the steps to find the angle of a sector from its arc length?

e.g. arc length $4\pi$ cm, radius $8$ cm

1. Full circumference: $2\pi \times 8 = 16\pi$ cm

2. Fraction of the circle: $4\pi \div 16\pi = \frac{1}{4}$

3. Multiply by $360$: angle $= 90^\circ$

True or false? A sector with an angle of $60^\circ$ has an arc length equal to $\frac{1}{6}$ of the circumference.

True. $\frac{60}{360} = \frac{1}{6}$ of the whole circle.

What are the steps to find the area of a shape made from a rectangle with a semicircle on one end?
e.g. rectangle $10$ cm by $6$ cm, with a semicircle of diameter $6$ cm on one end

1. Area of the rectangle: $10 \times 6 = 60$ cm$^2$
2. Area of the semicircle: $\frac{1}{2} \times \pi \times 3^2 = 14.13...$ cm$^2$
3. Add the parts: total area $= 74.1$ cm$^2$ (3 s.f.)

True or false? To find the perimeter of a shape made from a rectangle with a semicircle joined to one end, you include the straight edge where the two shapes meet.

False. That edge is inside the shape, so the perimeter is the three outer sides of the rectangle plus the semicircle's arc.