Arc Length and Sector Area

You're viewing free preview questions — 4 questions previewing here — subscribe to unlock 20 more, plus the Full Question Bank, Mock Papers & Worksheets.Upgrade

Arc Length and Sector Area — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionAngles at the Centre and in the CircleAssessment Practice

5 marks~8 minCriterion C

The diagram shows a circle with centre

O

. Points

A

B

C

and

D

lie on the circle.

C

lies on the major arc

AB

and

D

lies on the minor arc

AB

. The angle

AOB = 80°

Write down the size of angle

ACB

. [1]

Find the size of angle

ADB

. Show all working. [2]

A student claims that angles

ACB

and

ADB

are supplementary. Justify whether this claim is correct. [2]

Solutions

2QuestionAngles at the Centre and in the CircleAssessment Practice

6 marks~9 minCriterion A

The diagram shows a circle with centre

O

. Points

A

B

, and

C

lie on the circumference, with

C

on the major arc

AB

. The central angle

\angle AOB = 112°

Write down the value of

\angle OBA

, giving a reason. [1]

Find the value of

\angle OAB

. [2]

A fourth point

D

lies on the minor arc

AB

. The angle

\angle ADB = y°

. Justify the value of

y

, using two circle theorems. [3]

Solutions

3QuestionLength of arc and chord, perimeter and area of sector and segmentAssessment Practice

8 marks~12 minCriterion B

The table shows arc length and sector area values for four sectors, each from a circle of radius 10 cm.

Central angle (degrees)	30	60	90	120
Arc length (cm)	5.24	10.47	15.71	20.94
Sector area (cm²)	26.18	52.36	78.54	104.72

Describe the relationship between the central angle

\theta

and the arc length

L

, referring to at least two values in the table. [2]

Using the pattern in the table, write down a formula for the arc length

L

of a sector with central angle

\theta

degrees and radius

r

cm. Hence find the arc length of a sector with central angle

150^\circ

in the same circle (radius 10 cm). Give your answer in exact form and correct to 1 decimal place. [3]

A different circle has radius

r

cm. Two sectors of this circle have central angles

\alpha

degrees and

\beta

degrees respectively, where

\alpha + \beta = 180

. Show that the sum of their arc lengths equals

\pi r

, and justify why this result holds regardless of the individual values of

\alpha

and

\beta

. [3]

Solutions

4QuestionLength of arc and chord, perimeter and area of sector and segmentAssessment Practice

10 marks~15 minCriterion D

A city planner is designing a grass-covered traffic roundabout. Each roundabout is modelled as a sector of a circle with central angle

60°

. The diagram shows a sector with two radii of length

r

metres and the arc.

Sod (grass turf) costs 4.50 USD per square metre. The sector area formula is

A = \frac{\pi r^2}{6}.

The table below shows the installed radius and actual measured area at three sites.

Installed radius (m)	11.5	12.0	12.5
Actual area (m²)	68.00	75.00	82.50

Calculate the predicted area

A

for each installed radius using

A = \dfrac{\pi r^2}{6}

. Give each answer correct to 2 decimal places. [3]

Find the percentage error between the predicted and actual area at each site using

\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{\text{actual}} \times 100\%.

Hence determine the site at which the model is least accurate, and find the difference in sod cost between the predicted and actual amounts at that site. Give the cost difference in USD, correct to 2 decimal places. [4]

The planner proposes ordering sod based on the model prediction for

r = 12.0

m and adding a 10% buffer for waste. Advise the planner whether this strategy is sufficient to cover the actual area at every site in the table, and identify one limitation of using this sector model for a real roundabout installation. [3]

Solutions

Questions 5-24

Unlock these questions with Premium

That's every free question here — 4 of 24.

+20 more questions, plus Mock Papers & Worksheets, are waiting in Premium.

Upgrade