ESSENTIAL MATHEMATICS 1
WEEK 12 NOTES AND EXERCISES
Circles are popular shapes for formal gardens, fountains and pools.
Parts of a Circle
The diameter of a circle is a line that goes from one side of a circle to the other, through the centre. It is
always twice as long as the radius, which goes from the centre of the circle to the circular edge.
The perimeter, or the distance around the outside, of a circle is called the circumference. The formula for the
circumference of a circle is:
C = π × d or
=2×π×r
where d is the diameter and r is the radius.
π is the ratio of the circumference to the diameter of a circle. Its decimal value is 3.141592654………..
Example
Calculate the circumference of this circle, correct to one decimal place.
Solution
r = 8 cm and C = 2 × 𝜋 × r
=2×π×8
= 50.265 cm
= 50.3 cm, to one decimal place.
Example
Find the total perimeter of the quarter-circle correct to 1 decimal place.
Solution
1
90
1
The arc AB is 4 of the circumference of the circle because the angle is 90˚ and 360 = 4
C =2×𝜋×r
=2×𝜋×6
= 37.699
1
37.699 × 4 = 9.4 cm to one decimal place.
The total perimeter becomes 6 + 6 + 9.4 = 21.4 cm
Note
For a semi-circle divide by 2
When the angle is 90˚ divide by 4
When the angle is 60˚ divide by 6
When the angle is 45˚ divide by 8
Exercise Set 1
Q1. Calculate the perimeter of each figure. Answer correct to 1 decimal place.
a)
b)
c)
Q2. What fraction of the circumference of each circle is the red arc?
a)
b)
c)
Q3. Calculate the total perimeter of each of these shapes. Answer correct to 1 decimal place.
a)
d)
b)
e)
c)
f)
Q4. The running track around the outside of a soccer field is in the shape of a rectangle with a semicircle at
each end. How far will Katelyn travel when she runs once around the track?
Q5. a) A bicycle wheel has a radius of 33 cm. How far will the bike travel when the wheel rotates once?
Answer in metres correct to 1 decimal place.
b) The organisers of the Jimmy’s Beach Charity Marathon wanted to check the length of the course.
One of the organisers rode a bike, with wheels of radius 33 cm, from the start to the finish line. The bike was
equipped with a meter to count the number of times the wheel rotated. The meter showed 4824 rotations.
How long was the Charity Marathon Course?
Q6. The deck in Narelle’s house overlooks her garden, The diagram shows the shape of the deck.
a) What is the diameter of the semicircular section of the deck?
b) Calculate the length of the semicircular edge of the deck (i.e. the circular edge from A to B).
Answer in metres correct to 2 decimal places.
c) What is the perimeter of the deck? Answer in metres correct to 2 decimal places.
d) Narelle needs to replace the railing on the section of deck from C, through B to A. The railing is going to
cost $58 per metre. Calculate the total cost of replacing the railing.
Q7. The diagram shows the driveway to Deanna’s house, which is in the shape of a semicircle. How much
longer is the longer side of the driveway than the shorter side?
Area of a Circle
The formula for the area of a circle is:
A = π × r2
(which is the same as π × r × r)
Example
The Lightning Ridge mineral water baths are circular. The diameter of the baths is 12 m. Calculate the area
of the top of the water, correct to the nearest square metre.
Solution
The diameter is 12 m.
Radius 12 ÷2 m
6 m
Area π×r
π×π×36 m2
= 113 m2 to the nearest m2
Exercise Set 2
Q1. Find the areas of these circles, correct to 1 decimal place.
a)
b)
c)
Q2. Calculate the areas of these sectors, correct to 1 decimal place.
a)
b)
c)
Q3. The diameter of an ornamental water lily pond is 8 m. What is the area of the top of the pond?
Q4. Find the size of the shaded area in each of these figures. Answer correct to the nearest square
centimetre.
a)
b)
5. The shaded section in this figure is called an annulus.
a) Calculate the area of the smaller circle, which has a radius of 8 cm. Answer correct to 1 decimal place.
b) What is the radius of the larger circle?
c) Calculate the area of the larger circle, correct to 1 decimal place.
d) Hence find the size of the shaded area, correct to the nearest square centimetre.
Sketching Prisms
These solids are all prisms. Their cross-sections are shaded. The cross-section of a prism is the same all the
way through. It’s easy to sketch prisms if you follow a few easy steps and use a ruler.
Step 1: Draw a plane shape; for example, a triangle.
Step 2: Draw an identical shape across and up from the first shape. Keep matching lines parallel.
Step 3: Join the matching angles with straight lines. Use dashed lines for edges that you can’t actually see
from outside.
Exercise Set 3
Q1. Follow the three steps above to construct prisms with these cross-sections. Draw them on a blank sheet
of paper (from Steve).
Q2. Draw three more prisms with these cross-sections. (on blank paper)
Q3. Match the prisms in questions 1 to 8 with their cross-sections in A to H at the
bottom of the page.