Define an arc of a circle.

An arc is a part of the circumference of a circle. Two points on the circumference create two arcs, the smaller called the minor arc and the larger the major arc.

Define a sector of a circle.

A sector is the part of a circle enclosed by two radii and an arc, like a slice of a circular pizza. The curved edge of a sector is an arc, and two radii create both a minor and a major sector.

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Circle Geometry (SQA National 5 Maths): Flashcards

Exam code: X847 75

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Define an arc of a circle.

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Cards in this collection (13)

Define an arc of a circle.
An arc is a part of the circumference of a circle.
Two points on the circumference create two arcs, the smaller called the minor arc and the larger the major arc.
Complete the formula for the length of an arc.
$arc length = \frac{______}{360} \times______$
The completed formula is:
$arc length = \frac{θ}{360} \times 2 π r$
The fraction is the share of the full turn taken up by the angle at the centre, and $2 π r$ is the whole circumference.
True or False?
The formulae for the circumference and the area of a circle are given to you on the Formulae List.
False.
Neither $C = 2 π r$ nor $A = π r^{2}$ appears on the Formulae List, so both have to be remembered.
They carry forward from National 4 and are needed for every arc and sector calculation.
In the arc length formula why is the angle divided by $360$ first?
A full turn is $360 °$ , so that fraction is the share of the whole circle the sector takes up.
Multiplying the whole circumference by it leaves just the part that forms the arc.
Define a sector of a circle.
A sector is the part of a circle enclosed by two radii and an arc, like a slice of a circular pizza.
The curved edge of a sector is an arc, and two radii create both a minor and a major sector.
A sector has radius $21$ cm and an angle of $150 °$ so how is its area found?
Use $sector area = \frac{θ}{360} \times π r^{2}$ , which here gives $\frac{150}{360} \times π \times 21^{2}$ .
That works out as $577.3 {cm}^{2}$ to one decimal place.
How do you find the length of a major arc?
Use the same formula, but with the reflex angle at the centre, since that is the angle the major arc turns through.
For a radius of $24$ cm and an angle of $210 °$ that gives $\frac{210}{360} \times 2 π \times 24 = 88.0 cm$ to one decimal place.
Which angle does $θ$ stand for in the arc and sector formulae?
$θ$ is the angle at the centre of the circle, formed by the two radii meeting the ends of the arc.
On a diagram it is the angle inside the sector, not the reflex angle outside it, unless the question is about the major arc or sector.
What do you do when the arc length is given and the angle is not?
Start with the formula containing the two things you already know, substitute them in, and solve the equation for the one you want.
Knowing the arc length and the radius lets you find $θ$ , which can then be used to work out the sector area.
True or False?
Knowing the radius and the arc length is enough to find the angle at the centre.
True.
The arc length formula links the arc, the radius and the angle, so knowing any two of them gives the third.
Substituting an arc of $24$ cm and a radius of $18$ cm leaves an equation in which $θ$ is the only unknown.
Why is it worth keeping an exact value in terms of $π$ rather than rounding?
Rounding partway through introduces an error that then grows through the rest of the working.
The $π$ often cancels at a later step as well, so keeping it exact can make the arithmetic simpler rather than harder.
A sector has radius $18$ cm and arc length $24$ cm so how is its area found?
Solve $24 = \frac{θ}{360} \times 2 π \times 18$ to get $θ = \frac{240}{π}$ , and keep that exact.
Substituting it into the sector area formula gives $\frac{2}{3} \times 18^{2} = 216 {cm}^{2}$ , because the $π$ cancels.
What units does a sector area answer take?
Square units, so a radius measured in centimetres gives an area in ${cm}^{2}$ .
An arc length is a distance rather than an area, so it takes plain $cm$ instead.

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