The CENTRE for EDUCATION in
MATHEMATICS and COMPUTING
MEASUREMENT: CIRCLES
This resource may be copied in its entirety, but is not to be used for commercial purposes without permission from the Centre for Education in
Mathematics and Computing, University of Waterloo.
Play the Basketball Geometry game first!
Click on http://www.factmonster.com/math/knowledgebox/player.html?movie=sfw41551 or
go to www.wiredmath.ca for the link.
1.
2.
a. Draw each circle on graph paper.
i. radius 1 cm
ii. diameter 6 cm
iii. diameter 5 cm
b. Estimate the circumference of each circle?
c. Estimate the area of each circle.
iv. radius 2 cm
a. Using the results from Question 1b, how many times greater is the circumference than the diameter?
b. Using the results from Question 1b, how many times greater is the circumference than the radius?
The ratio of the circumference to the radius is the same for any circle; that is,
C
r
= 2π ,
where π is a constant. The diameter of a circle is two times the length of its radius.
Therefore, the ratio of the circumference to the diameter is always π; that is,
Greek letter
π is pronounced “pie” and spelled “pi”.
C
d
= π . The
π = 3.141592653589… . The digits of pi go on forever. It is nonrepeating.
* Use π = 3.14 for all the questions in this worksheet.
3.
Given that
C
d
= π , where C stands for circumference and d for diameter, write a formula that can be
used to calculate the circumference when the diameter is known.
4.
Calculate the circumference of each circle, correct to two decimal places.
a.
b.
Expectation: solve problems the estimation and calculation of the circumference and area of a circle. For more activities and resources from the
University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca.
1
5.
Determine the perimeter of the
outer frame of the window.
6.
Calculate the perimeter of each figure.
1 cm
1m
b.
a.
1 cm
c.
d.
2m
Area of a Circle
The circle is cut in half, and
each half is divided into four
equal segments.
r
r
As such division gets finer and
finer, the segments fit
together to form a figure that
is roughly a parallelogram.
b
The base of the parallelogram is formed by half the circumference of the circle.
The area of circle is A= b × r
=
=
1
2
1
2
×C ×r
× 2×π × r × r
= π × r2
The formula for the area of a circle is
7.
(since Cπr
=2
)
A = πr2 .
Calculate the area of each circle that has:
a. radius of 5 cm.
b. radius of 4.7 m.
c. diameter of 8 cm.
d. diameter of 3.2 cm.
Expectation: solve problems the estimation and calculation of the circumference and area of a circle. For more activities and resources from the
University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca.
2
8.
Complete the table.
Radius
7 cm
Circumference and Area of Circles
Circumference
Area
21.98 m
28.26 cm 2
9.
The hoot of a lost African wild dog can be heard by people from as far away as 4 kilometres. What is
the area in which people can hear a hoot?
10.
Calculate the area of the shaded region.
a.
b.
c.
1 cm
8m
3 cm
6 cm
1 cm
11.
Typhoons are classified by their size and strength. The minimum radius of a “very large” typhoon is
800 km. What is the smallest possible area covered by a very large typhoon?
12.
In the diagram, circular arcs PQ, QR, and ST have centres P
T, S, and Q respectively. If PT equals one metre, then
determine the perimeter of figure PQRST?
Q
R
60º
60º
T
60º
S
Don’t forget now! Go to www.wiredmath.ca for the link.
TRY THESE!
Calculate the circumference of a circle
http://www.aaamath.com/B/g59_cix1.htm#section2
Determine the area of a circle
http://www.aaamath.com/B/g5_612x1.htm#section2
Expectation: solve problems the estimation and calculation of the circumference and area of a circle. For more activities and resources from the
University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca.
3
CHALLENGE YOURSELF
13.
AB is the diameter of the semicircle shown. If AC = 8 cm, CB = 6 cm,
and ∠ACB = 90º, what is the area of the shaded portion?
C
A
14.
B
A rectangular house that measures 20 m by 10 m has an outside electrical outlet at a corner of the
house. An electric mower, connected by a cord to the outlet, can reach a maximum distance of 15 m.
What is the largest area of lawn that can be cut?
EXTENSIONS
15.
A square with edge length 2 cm has semicircles drawn on each side.
Find the total area of the shaded region.
16.
A
c
B
ABC is a right-angled triangle with vertex A on the
semicircle drawn, with a as diameter. Semicircles are also
constructed, as shown, with diameters b and c.
What is the area of the shaded region?
b
a
C
bc
a.
2
π b2 π c2 π a2
c.
−
+
4
4
4
2
π a bc
e.
−
4
2
π b2
π c2
π a2
+
−
2
2
2
2
2
π b π c π a2
d.
+
−
8
8
8
b.
Expectation: solve problems the estimation and calculation of the circumference and area of a circle. For more activities and resources from the
University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca.
4