num-title-ct-core

Circles, SM-Bank 040

Use the arc length formula below to calculate the arc length of the sector, correct to one decimal place. (2 marks)

\(l=\dfrac{\theta}{360}\times 2\pi r\)

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\(25.0\ \text{m (1.d.p.)}\)

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\(l\)	\(=\dfrac{\theta}{360}\times 2\pi r\)
	\(=\dfrac{135}{360}\times 2\pi \times 10.6\)
	\(=24.9756\dots\)
	\(=25.0\ \text{m (1.d.p.)}\)

Circles, SM-Bank 039

Use the arc length formula below to calculate the arc length of the sector, correct to one decimal place. (2 marks)

\(l=\dfrac{\theta}{360}\times 2\pi r\)

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\(4.2\ \text{cm (1.d.p.)}\)

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\(l\)	\(=\dfrac{\theta}{360}\times 2\pi r\)
	\(=\dfrac{30}{360}\times 2\pi \times 8\)
	\(=4.1887\dots\)
	\(=4.2\ \text{cm (1.d.p.)}\)

Circles, SM-Bank 038

Evan rides his bike around a circular track with a radius of 150 metres.

He rides the track 5 days a week and wants to have ridden 230 kilometres by the end of the week.

What is the minimum number of laps, to the nearest whole lap, he must complete each day to achieve his target? (3 marks)

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\(49\ \text{laps}\)

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\(\text{Total Distance}=230\ \text{km}=230\ 000\ \text{m}\)

\(\text{Total laps}\)	\(=\dfrac{230\ 000}{C}\)
	\(=\dfrac{230\ 000}{300\pi}\)
	\(=244.037\dots\)

\(\text{Laps per day:}\)	\(=\dfrac{244.037\dots}{5}\)
	\(=48.807\dots\)
	\(\approx 49\ \text{laps}\)

\(\therefore\ \text{Evan needs to ride }49\ \text{laps per day to achieve his goal}.\)

\(\Big[\text{Check: }49\times 5\times 300\pi=230\ 907.06\ \text{m}=230.907\ \text{km}\Big]\)

Circles, SM-Bank 037 MC

A car tyre has a circumference of 120 cm.

Which of these is closest to the radius of the tyre?

\(9.5\ \text{cm}\)
\(19.1\ \text{cm}\)
\(38.0\ \text{cm}\)
\(753.6\ \text{cm}\)

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\(B\)

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\(C\)	\(=2 \pi r\)
\(120\)	\(=2\times \pi\times r\)
\(r\)	\(=\dfrac{120}{2 \pi}\)
	\(=19.098\dots\)
	\(\approx 19.1\ \text{cm}\)

\(\Rightarrow B\)

Circles, SM-Bank 036 MC

A yoyo at the end of a string moves through a quarter circle arc, as shown below.

The string is 70 cm long.

The curved distance the yoyo has travelled is closest to

\(80\ \text{cm}\)
\(90\ \text{cm}\)
\(100\ \text{cm}\)
\(110\ \text{cm}\)

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\(D\)

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\(\text{Distance}\)	\(=\dfrac{1}{4}\times 2\pi r\)
	\(=\dfrac{1}{4}\times 2\pi 70\)
	\(= 109.9557\dots\)
	\(\approx 110\ \text{cm}\)

\(\Rightarrow D\)

Circles, SM-Bank 035

Guerra slings a fire torch in a circle at the end of a 3 metre rope.

How far has the fire torch travelled after completing 15 circles?

Give your answer correct to the nearest whole number. (2 marks)

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\(283\ \text{metres (nearest whole number)}\)

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\(\text{Distance travelled in 1 revolution:}\)

\(C\)	\(=2\pi r\)
	\(=2\times \pi\times 3\)
	\(= 18.8495\dots\)

\(\therefore\ \text{Distance travelled in }15\ \text{revolutions:}\)

\(=15\times 18.8495\dots\)

\(= 282.7433\dots\)

\(=283\ \text{metres (nearest whole number)}\)

Circles, SM-Bank 034 MC

A wheel on a bicycle has a diameter of 530 mm.

The wheel turns 12 times.

Approximately how many metres has the bicycle travelled?

\(6\ \text{m}\)
\(20\ \text{m}\)
\(40\ \text{m}\)
\(399\ \text{m}\)

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\(B\)

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\(\text{Diameter}=530\ \text{mm}=0.530\ \text{m}\)

\(\therefore\ \text{Distance}\)	\(=12\times \text{circumference}\)
	\(=12\times \pi d\)
	\(=12\times \pi\times 0.530\)
	\(=19.9805\dots\)
	\(=20\ \text{m (nearest m)}\)

\(\Rightarrow B\)

Circles, SM-Bank 021

Percy has a large shower head which has a diameter of 25.5 centimetres.

What is the circumference of the shower head in centimetres, correct to one decimal place? (2 marks)

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\(80.1\ \text{cm}\ (1\ \text{d.p.})\)

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\(C\)	\(=\pi d\)
	\(=\pi\times 25.5\)
	\(=80.110\dots\)
	\(=80.1\ \text{cm}\ (1\ \text{d.p.})\)

Circles, SM-Bank 033

Calculate the radius of a circle with a circumference of 870 metres, correct to the nearest ten metres. (2 marks)

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\(140\ \text{m}\ (\text{nearest ten})\)

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\(C\)	\(=2\pi r\)
\(870\)	\(=2\pi r\)
\(r\)	\(=\dfrac{870}{2\pi}\)
	\(=138.4648\dots\)
	\(=140\ \text{m}\ (\text{nearest ten})\)

Circles, SM-Bank 032

Calculate the radius of a circle with a circumference of 324 centimetres, correct to 2 decimal places. (2 marks)

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\(51.57\ \text{cm}\ (2\ \text{d.p.})\)

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\(C\)	\(=2\pi r\)
\(324\)	\(=2\pi r\)
\(r\)	\(=\dfrac{324}{2\pi}\)
	\(=51.5662\dots\)
	\(=51.57\ \text{cm}\ (2\ \text{d.p.})\)

Circles, SM-Bank 031

Calculate the diameter of a circle with a circumference of 109.8 metres, correct to 2 decimal places. (2 marks)

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\(34.95\ \text{m}\ (2\ \text{d.p.})\)

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\(C\)	\(=\pi d\)
\(109.8\)	\(=\pi d\)
\(d\)	\(=\dfrac{109.8}{\pi}\)
	\(=34.9504\dots\)
	\(=34.95\ \text{m}\ (2\ \text{d.p.})\)

Circles, SM-Bank 030

Calculate the diameter of a circle with a circumference of 24 centimetres, correct to 1 decimal place. (2 marks)

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\(7.6\ \text{cm}\ (1\ \text{d.p.})\)

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\(C\)	\(=\pi d\)
\(24\)	\(=\pi d\)
\(d\)	\(=\dfrac{24}{\pi}\)
	\(=7.639\dots\)
	\(=7.6\ \text{cm}\ (1\ \text{d.p.})\)

Circles, SM-Bank 029

Calculate the circumference of a circle with a radius of 1012 centimetres, correct to the nearest hundred. (2 marks)

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\(6400\ \text{cm}\ (\text{nearest hundred})\)

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\(C\)	\(=2\pi r\)
	\(=2\pi\times 1012\)
	\(=6358.5835\dots\)
	\(=6400\ \text{cm}\ (\text{nearest hundred})\)

Circles, SM-Bank 028

Calculate the circumference of a circle with a radius of 453 kilometres, correct to the nearest whole number. (2 marks)

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\(2846\ \text{km}\ (\text{nearest whole number})\)

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\(C\)	\(=2\pi r\)
	\(=2\pi\times 453\)
	\(=2846.2829\dots\)
	\(=2846\ \text{km}\ (\text{nearest whole number})\)

Circles, SM-Bank 027

Calculate the circumference of a circle with a diameter of 24.6 millimetres, correct to 0ne decimal place. (2 marks)

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\(77.3\ \text{mm}\ \ (1\ \text{d.p.})\)

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\(C\)	\(=\pi d\)
	\(=\pi\times 24.6\)
	\(=77.2831\dots\)
	\(=77.3\ \text{mm}\ \ (1\ \text{d.p.})\)

Circles, SM-Bank 026

Calculate the circumference of a circle with a diameter of 35 centimetres, correct to two decimal places. (2 marks)

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\(109.96\ \text{cm}\ \ (2\ \text{d.p.})\)

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\(C\)	\(=\pi d\)
	\(=\pi\times 35\)
	\(=109.9557\dots\)
	\(=109.96\ \text{cm}\ \ (2\ \text{d.p.})\)

Circles, SM-Bank 025

Calculate the circumference of the circle below correct to two decimal places. (2 marks)

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\(653.45\ \text{cm}\ \ (2\ \text{d.p.})\)

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\(C\)	\(=2\pi r\)
	\(=2\pi\times 104\)
	\(=653.4512\dots\)
	\(=653.45\ \text{cm}\ \ (2\ \text{d.p.})\)

Circles, SM-Bank 024

Calculate the circumference of the circle below correct to one decimal place. (2 marks)

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\(44.0\ \text{cm}\ \ (1\ \text{d.p.})\)

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\(C\)	\(=2\pi r\)
	\(=2\pi\times 7\)
	\(=43.9822\dots\)
	\(=44.0\ \text{cm}\ \ (1\ \text{d.p.})\)

Circles, SM-Bank 023

Calculate the circumference of the circle below correct to one decimal place. (2 marks)

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\(38.3\ \text{m}\ \ (1\ \text{d.p.})\)

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\(C\)	\(=\pi d\)
	\(=\pi\times 12.2\)
	\(=38.3274\dots\)
	\(=38.3\ \text{m}\ \ (1\ \text{d.p.})\)

Circles, SM-Bank 022

Calculate the circumference of the circle below correct to two decimal places. (2 marks)

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\(15.71\ \text{m}\ \ (2\ \text{d.p.})\)

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\(C\)	\(=\pi d\)
	\(=\pi\times 5\)
	\(=15.7079\dots\)
	\(=15.71\ \text{m}\ \ (2\ \text{d.p.})\)

Circles, SM-Bank 020

Use a calculator to evaluate \(2\large\pi r\) correct to 1 decimal place, when \(\large r\)\(=2.1\). (2 marks)

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\(13.2\ (1\ \text{d.p.})\)

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\(2\pi r\)	\(=2\times \pi\times 2.1\)
	\(=13.194\dots\)
	\(=13.2\ (1\ \text{d.p.})\)

Circles, SM-Bank 019

Evaluate \(5\large\pi\) using a calculator, giving your answer correct to 2 decimal places. (1 mark)

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\(15.71\ (2\ \text{d.p.})\)

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\(5\pi\)	\(=15.7079\dots\)
	\(=15.71\ (2\ \text{d.p.})\)

Circles, SM-Bank 018 MC

Which of the following is the definition of the circumference of a circle?

The distance across a circle passing through the centre.
A portion of the circumference of a circle.
The distance along the boundary of the circle.
The distance from the centre of the circle to its circumference.

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\(C\)

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\(\text{The circumference is the distance along}\)

\(\text{the boundary a circle.}\)

\(\Rightarrow C\)

Circles, SM-Bank 017 MC

Which of the following is the definition of the radius of a circle?

The distance across a circle passing through the centre.
A portion of the circumference of a circle.
The distance around the circle.
The distance from the centre of the circle to its circumference.

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\(D\)

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\(\text{The radius is the distance from the centre of}\)

\(\text{ a circle to the circumference.}\)

\(\Rightarrow D\)

Circles, SM-Bank 016 MC

Which of the following is the definition of the diameter of a circle?

The distance across a circle passing through the centre.
A portion of the circumference of a circle.
The distance around the circle.
The distance from the centre of the circle to its circumference.

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\(A\)

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\(\text{The diameter is the distance across a circle}\)

\(\text{passing through the centre.}\)

\(\Rightarrow A\)

Circles, SM-Bank 015 MC

Which of the following is the definition of an arc?

The distance across a circle passing through the centre.
A portion of the circumference of a circle.
The distance around the circle.
A line that touches a circle at only one point.

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\(B\)

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\(\text{An arc is a section of the circumference of a circle.}\)

\(\Rightarrow B\)

Circles, SM-Bank 014

What fraction of a circle is indicated in the diagram below? (1 mark)

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\(\dfrac{1}{12}\)

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\(\text{Angle}\)	\(=\dfrac{\theta}{360}\)
	\(=\dfrac{30}{360}\)
	\(=\dfrac{1}{12}\)

Circles, SM-Bank 013

What fraction of a circle is indicated in the diagram below? (1 mark)

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\(\dfrac{1}{4}\)

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\(\text{Angle}\)	\(=\dfrac{\theta}{360}\)
	\(=\dfrac{90}{360}\)
	\(=\dfrac{1}{4}\)

Circles, SM-Bank 012

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What fraction of a circle is indicated in the diagram below? (1 mark)

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\(\dfrac{3}{8}\)

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\(\text{Angle}\)	\(=\dfrac{\theta}{360}\)
	\(=\dfrac{135}{360}\)
	\(=\dfrac{3}{8}\)

Circles, SM-Bank 011

What fraction of a circle is indicated in the diagram below? (1 mark)

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\(\dfrac{1}{8}\)

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\(\text{Angle}\)	\(=\dfrac{\theta}{360}\)
	\(=\dfrac{45}{360}\)
	\(=\dfrac{1}{8}\)

Circles, SM-Bank 010

What fraction of a circle is indicated in the diagram below? (1 mark)

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\(\dfrac{1}{3}\)

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\(\text{Angle}\)	\(=\dfrac{\theta}{360}\)
	\(=\dfrac{120}{360}\)
	\(=\dfrac{1}{3}\)

Circles, SM-Bank 009

What fraction of a circle is indicated in the diagram below? (1 mark)

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\(\dfrac{1}{6}\)

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\(\text{Angle}\)	\(=\dfrac{\theta}{360}\)
	\(=\dfrac{60}{360}\)
	\(=\dfrac{1}{6}\)

Circles, SM-Bank 008

If the diameter of a circle is 16.2 millimetres, what is the length of its radius? (1 mark)

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\(8.1\ \text{mm}\)

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\(\text{Radius}\)	\(=\dfrac{\text{diameter}}{2}\)
	\(=\dfrac{16.2}{2}\)
	\(=8.1 \text{mm}\)

Circles, SM-Bank 007

If the radius of a circle is 24 centimetres, what is the length of its diameter? (1 mark)

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\(48\ \text{cm}\)

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\(\text{Diameter}\)	\(=2\times\text{radius}\)
	\(=2\times 24\)
	\(=48\ \text{cm}\)

Circles, SM-Bank 006

Name the circle parts indicated in the drawing below. (2 marks)

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a. \(\text{Secant}\)

b. \(\text{Quadrant}\)

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a. \(\text{Secant}\)

b. \(\text{Quadrant}\)

Circles, SM-Bank 005

Name the circle parts indicated in the drawing below. (2 marks)

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a. \(\text{Major segment}\)

b. \(\text{Tangent}\)

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a. \(\text{Major segment}\)

b. \(\text{Tangent}\)

Circles, SM-Bank 004

Name the circle parts indicated in the drawing below. (2 marks)

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a. \(\text{Centre}\)

b. \(\text{Arc}\)

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a. \(\text{Centre}\)

b. \(\text{Arc}\)

Circles, SM-Bank 003

Name the circle parts indicated in the drawing below. (2 marks)

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a. \(\text{Semicircle}\)

b. \(\text{Minor segment}\)

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a. \(\text{Semicircle}\)

b. \(\text{Minor segment}\)

Circles, SM-Bank 002

Name the circle parts indicated in the drawing below. (2 marks)

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a. \(\text{Radius}\)

b. \(\text{Sector}\)

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a. \(\text{Radius}\)

b. \(\text{Sector}\)

Circles, SM-Bank 001

Name the circle parts indicated in the drawing below. (2 marks)

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a. \(\text{Diameter}\)

b. \(\text{Chord}\)

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a. \(\text{Diameter}\)

b. \(\text{Chord}\)

Rectilinear Figures, SM-Bank 045

An isosceles triangle has a base of length 4 centimetres and a perimeter of 20 centimetres.

Find the length of the equal sides. (2 marks)

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\(8\ \text{cm}\)

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\(\text{Let the length of the equal sides be}\ x\)

\(\text{Then,}\ \ 2\times x+4\)	\(=20\)
\(2x\)	\(=20-4\)
\(2x\)	\(=16\)
\(x\)	\(=8\)

\(\therefore\ \text{The equal sides are of length }8\ \text{cm}\)

Rectilinear Figures, SM-Bank 044

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Find the perimeter of the following irregular shape. (2 marks)

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\(70\ \text{mm}\)

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\(\text{Perimeter}\)

\(=7+13+20+9+6+15\)

\(=70\ \text{mm}\)

Rectilinear Figures, SM-Bank 043

Find the perimeter of the following irregular shape. (2 marks)

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\(38\ \text{cm}\)

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\(\text{Perimeter}\)

\(=10+5+3+4+7+9\)

\(=38\ \text{cm}\)

Rectilinear Figures, SM-Bank 042

Find the perimeter of the following irregular shape. (2 marks)

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\(38.8\ \text{m}\)

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\(\text{Perimeter}\)

\(=10.2+2\times 3.1 +4\times 2.4+4\times 1.8+5.6\)

\(=38.8\ \text{m}\)

Rectilinear Figures, SM-Bank 041

Mark all missing lengths on the following shape. (1 mark)

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Calculate the perimeter. (2 marks)

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b. \(390\ \text{m}\)

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b. \(\text{Perimeter}\)

\(=87+5\times 32+44+99\)

\(=390\ \text{m}\)

Rectilinear Figures, SM-Bank 040

Mark all missing lengths on the following shape. (1 mark)

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Calculate the perimeter. (2 marks)

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b. \(54\ \text{mm}\)

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b. \(\text{Perimeter}\)

\(=13+2\times 9+4+2\times 5+3+6\)

\(=54\ \text{mm}\)

Rectilinear Figures, SM-Bank 039

Mark all missing lengths on the following shape. (1 mark)

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Calculate the perimeter. (2 marks)

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b. \(52\ \text{cm}\)

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b. \(\text{Perimeter}\)

\(=12+14+5+10+7+4\)

\(=52\ \text{cm}\)

Rectilinear Figures, SM-Bank 038

Calculate the perimeter of the following polygon. (2 marks)

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\(6\ \text{m}\)

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\(\text{Perimeter}\)

\(=5\times 1.2\)

\(=6\ \text{m}\)

Rectilinear Figures, SM-Bank 037

Calculate the perimeter of the following trapezium. (2 marks)

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\(28.8\ \text{mm}\)

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\(\text{Perimeter}\)

\(=6+7.2+4.8+10.8\)

\(=28.8\ \text{mm}\)

Rectilinear Figures, SM-Bank 036

Calculate the perimeter of the following kite. (2 marks)

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\(34.2\ \text{cm}\)

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\(\text{Perimeter}\)

\(=2\times 12.3+2\times 4.8\)

\(=34.2\ \text{cm}\)

Rectilinear Figures, SM-Bank 035

Calculate the perimeter of the following parallelogram. (2 marks)

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\(228\ \text{cm}\)

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\(\text{Perimeter}\)

\(=2\times 73+2\times 41\)

\(=228\ \text{cm}\)

Rectilinear Figures, SM-Bank 034

Calculate the perimeter of the following triangle. (2 marks)

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\(38.8\ \text{m}\)

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\(\text{Perimeter}\)

\(=15.3+12+11.5\)

\(=38.8\ \text{m}\)

Rectilinear Figures, SM-Bank 033

Calculate the perimeter of the following triangle. (2 marks)

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\(15.3\ \text{mm}\)

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\(\text{Perimeter}\)

\(=3\times 5.1\)

\(=15.3\ \text{mm}\)

Rectilinear Figures, SM-Bank 032

Calculate the perimeter of the following triangle. (2 marks)

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\(51\ \text{cm}\)

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\(\text{Perimeter}\)

\(=2\times 19+13\)

\(=51\ \text{cm}\)

Rectilinear Figures, SM-Bank 031

Calculate the perimeter of the following rhombus. (2 marks)

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\(64\ \text{cm}\)

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\(\text{Perimeter}\)

\(=4\times 16\)

\(=64\ \text{cm}\)

Rectilinear Figures, SM-Bank 030

Calculate the perimeter of the following square. (2 marks)

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\(42\ \text{m}\)

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\(\text{Perimeter}\)

\(=4\times 10.5\)

\(=42\ \text{m}\)

Rectilinear Figures, SM-Bank 029

Calculate the perimeter of the following rectangle. (2 marks)

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\(366\ \text{mm}\)

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\(\text{Perimeter}\)

\(=2\times 36 +2\times 147\)

\(=366\ \text{mm}\)

Rectilinear Figures, SM-Bank 028

Calculate the perimeter of the following rectangle. (2 marks)

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\(35.8\ \text{cm}\)

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\(\text{Perimeter}\)

\(=2\times 12.3 +2\times 5.6\)

\(=35.8\ \text{cm}\)

Rectilinear Figures, SM-Bank 027

A farmer uses the river bordering his property and fencing to create a large grazing paddock for his horses, as shown in the diagram below.

The fencing has 4-strand wiring which is also shown below.

Given that fencing is not required on the river boundary, how many kilometres of wire will be required to fence the rest of the paddock? (2 marks)

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\(5.6\ \text{km}\)

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\(\text{Wire needed}\)

\(=4\times(700+200+300+1100+200+200+1200)\)

\(=4\times 3900\)

\(=15\ 600\ \text{m}=15.6\ \text{km}\)

Rectilinear Figures, SM-Bank 026

Jill cut a piece of binding in half.

She then placed one half on top of the other half, as shown below.

What is the perimeter of the newly formed shape in millimetres? (2 marks)

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\(1600\ \text{mm}\)

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\(\text{Calculating the perimeter clockwise from top right:}\)

\(\text{Perimeter}\)

\(=18+2\times 382 +18+2\times 400\)

\(=1600\ \text{mm}\)