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Find Angles Using Knowledge of Sectors

In this worksheet, students will use their knowledge of sectors to find angles when given an arc length and sector area.

'Find Angles Using Knowledge of Sectors ' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, OCR, Eduqas, Pearson Edexcel

Curriculum topic:   Geometry and Measures, Mensuration

Curriculum subtopic:   Mensuration and Calculation, Area Calculations

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

meat pie

 

So, you'd like the biggest piece of the pie, but have been told you can only have it if you can calculate the angle of it.

That is so unfair. The things you have to do!

 

As ever there is a formula to do this.

As we are looking at a part of a circle  is involved of course!

 

To find the area of a circle we use the formula Πr²  or (π x r x r)

 

It makes sense to use this to help us find the angle of a sector.

Remember you can use the π button on your calculator or use the value 3.14

 

Example

 

pie chart

 

The area of the dark blue sector of this pie chart is 25 cm²

The radius is 10 cm. Find the angle at the centre of this sector.

 

Angle = Area x 360 ÷ π x r x r

 25 x 360 = 9000

3.14 x 10 x 10 = 314

9000 ÷ 314 = 28.7° correct to 1 decimal place

 

Just follow the steps of the formula carefully and you can't go wrong!

 

In the following questions, 3.14 has been used as the value of π

Slice of pie

The area of this slice of pie is 20 cm²

The radius is 6 cm.

Find the angle at the centre of the pie.

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

57.3°

86.0°

63.7°

72.2°

Pie chart

 

The area of the light blue part of this pie chart is 75 cm²

The radius is 10 cm.

 

Find the angle at the centre of that sector of the pie chart.

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

 86.0°85.9°83.2°84.3°
The angle at the centre of the sector is

lattice pie

 

The area of the large sector of this pie chart is 95 cm²

The radius is 6 cm.

 

Find the angle at the centre of the large sector of the pie.

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

 86.0°85.9°83.2°84.3°
The angle at the centre of the sector is

Pineapple segment

 

The area of this sector of pineapple is 56 cm²

The radius is 5 cm.

 

Find the angle at the centre of the sector of pineapple.

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

 86.0°85.9°83.2°84.3°
The angle at the centre of the sector is

cake

A sector of this cake has is going to be cut with an area of 85 cm²

The radius is 8 cm.

 

Find the angle at the centre of the sector of cake once it has been cut.

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

153.2°

152.3°

142.3°

153.3°

Pie chart

 

The pink sector of this pie chart has an area of 162 cm²

The radius is 12 cm.

 

Find the angle at the centre of the sector. 

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

153.2°

152.3°

142.3°

153.3°

Blue cheese

 

The area of this sector of blue cheese is 15 cm²

The radius is 5 cm.

 

Find the angle at the centre of the sector of cheese

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

153.2°

152.3°

142.3°

153.3°

Pie chart

 

The area of the pie chart sector for nitrogen is 165 cm²

The radius is 8 cm.

 

Find the angle at the centre of that sector of the pie chart.

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

 229.3°259.6°269.5°295.6°
The angle at the centre of the sector is

 

This fan forms a sector of a circle.

The area of the fan is 27 cm²

The radius is 5 cm.

 

Find the angle at the centre of the fan.

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

 229.3°259.6°269.5°295.6°
The angle at the centre of the sector is

Can you match the angle of the sector to the correct measurements?

Column A

Column B

Area of sector 32 cm², Radius 4 cm
229.3°
Area of sector 12 cm², Radius 4 cm
229.3°
Area of sector 25 cm², Radius 4 cm
179.1°
Area of sector 288 cm², Radius 12 cm
86°
  • Question 1

Slice of pie

The area of this slice of pie is 20 cm²

The radius is 6 cm.

Find the angle at the centre of the pie.

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

CORRECT ANSWER
63.7°
EDDIE SAYS
That pie looks very tempting and delightful! Formulas are delightful (the hard part sometimes is remembering them)! If you apply the formula correctly you should get: 20 x 360 = 7200 3.14 x 6 x 6 = 113.04 Now 7200 ÷ 113.04 = 63.7° (correct to 1 d.p)
  • Question 2

Pie chart

 

The area of the light blue part of this pie chart is 75 cm²

The radius is 10 cm.

 

Find the angle at the centre of that sector of the pie chart.

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

CORRECT ANSWER
 86.0°85.9°83.2°84.3°
The angle at the centre of the sector is
EDDIE SAYS
Calculators vary slightly! Make sure you know how to use yours correctly. Here we've calculated the two parts of the formula separately before dividing. 75 x 360 = 27000 3.14 x 10 x 10 = 314 Now 27000 ÷ 314 = 85.98 (86.0° correct to 1 d.p)
  • Question 3

lattice pie

 

The area of the large sector of this pie chart is 95 cm²

The radius is 6 cm.

 

Find the angle at the centre of the large sector of the pie.

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

CORRECT ANSWER
EDDIE SAYS
Did you know that a slice of this pie costs $5.50 in Jamaica and $6.00 in Barbados? That is the pie rates of the Caribbean for you! 95 x 360 = 34200 3.14 x 6 x 6 = 113.04 Now 34200 ÷ 113.04 = 302.5° (correct to 1 d.p)
  • Question 4

Pineapple segment

 

The area of this sector of pineapple is 56 cm²

The radius is 5 cm.

 

Find the angle at the centre of the sector of pineapple.

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

CORRECT ANSWER
EDDIE SAYS
Why do pineapples love extreme sports? ... Because they are hardcore! Are you getting familiar with the formula yet? Just keep practising. 56 x 360 = 20160 3.14 x 5 x 5 = 78.5 Now 20160 ÷ 78.5 = 256.8° (correct to 1 d.p)
  • Question 5

cake

A sector of this cake has is going to be cut with an area of 85 cm²

The radius is 8 cm.

 

Find the angle at the centre of the sector of cake once it has been cut.

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

CORRECT ANSWER
152.3°
EDDIE SAYS
85 x 360 = 30600 3.14 x 8 x 8 = 200.96 Now 30600 ÷ 200.96 = 152.3° (correct to 1 d.p) Oh, that is a nice size piece of cake to go with a cup of tea!
  • Question 6

Pie chart

 

The pink sector of this pie chart has an area of 162 cm²

The radius is 12 cm.

 

Find the angle at the centre of the sector. 

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

CORRECT ANSWER
EDDIE SAYS
On questions such as this, they generally carry a lot of marks. Always show each stage of your working out to get as many marks as possible. 162 x 360 = 58320 3.14 x 12 x 12 = 452.16 58320 ÷ 452.16 = 129.0° (correct to 1 d.p)
  • Question 7

Blue cheese

 

The area of this sector of blue cheese is 15 cm²

The radius is 5 cm.

 

Find the angle at the centre of the sector of cheese

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

CORRECT ANSWER
EDDIE SAYS
How do ducks eat cheese? ... On quackers! Let's hope that these questions are not driving you crackers. As long as you are consistent in your working out, you'll find things will become easier. 15 x 360 = 5400 3.14 x 5 x 5 = 78.5 Now 5400 ÷ 78.5 = 68.8° (correct to 1 d.p)
  • Question 8

Pie chart

 

The area of the pie chart sector for nitrogen is 165 cm²

The radius is 8 cm.

 

Find the angle at the centre of that sector of the pie chart.

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

CORRECT ANSWER
 229.3°259.6°269.5°295.6°
The angle at the centre of the sector is
EDDIE SAYS
The same explanation again. That is the beauty of a lot of maths. You have a formula and just need to apply it! 165 x 360 = 59400 3.14 x 8 x 8 = 200.96 Now 59400 ÷ 200.96 = 295.6.° (correct to 1 d.p)
  • Question 9

 

This fan forms a sector of a circle.

The area of the fan is 27 cm²

The radius is 5 cm.

 

Find the angle at the centre of the fan.

 

Use 3.14 for the value of Π

Round your answer to 1 decimal place (1 d.p)

CORRECT ANSWER
EDDIE SAYS
Are you a fan of these now? 27 x 360 = 9720 3.14 x 5 x 5 = 78.5 Now 9720 ÷ 78.5 = 123.8° (correct to 1 d.p)
  • Question 10

Can you match the angle of the sector to the correct measurements?

CORRECT ANSWER

Column A

Column B

Area of sector 32 cm², Radiu...
229.3°
Area of sector 12 cm², Radiu...
86°
Area of sector 25 cm², Radiu...
179.1°
Area of sector 288 cm², Radi...
229.3°
EDDIE SAYS
Are you now feeling confident using this formula? Like anything, it is easy once you know, it just takes a bit of practice! That's another activity completed, let's try another to really consolidate our understanding!
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