Succeed with maths – Part 2
Succeed with maths – Part 2

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Succeed with maths – Part 2

3.3 Areas of circles

You can show that the area of a circle is calculated by multiplying the square of the radius by pi, by splitting the circle into equal sized segments and then arranging these to form a rectangle as shown in Figure 25:

Described image
Figure 25 Circle area by sectors

If each segment is made gradually smaller, any ‘bumps’ along the top and bottom edges will be smoothed out to form a line that is closer and closer to being straight. The height of the rectangle will then be the same as the radius of the circle and the length half of the circumference. The area of this rectangle will therefore be equivalent to that of the circle.

The area of a rectangle left parenthesis circle right parenthesis equals radius multiplication one divided by two multiplication circumference

The circumference and radius are related by the following formula:

Circumference equals two multiplication pi multiplication radius

So equation left hand side one divided by two multiplication circumference equals right hand side pi multiplication radius

So putting this together gives:

multiline equation line 1 Area of a circle equals radius multiplication pi multiplication radius line 2 equation left hand side equals right hand side pi multiplication radius squared

The formula for the area of a circle is therefore:

Area of a circle equals pi multiplication radius squared

If you are given the diameter of a circle instead of the radius, the first step to take when working out the area would be to halve the diameter.

See how you get on with applying this new formula in the next activity.

Activity 7 Everyday structures

Timing: Allow approximately 10 minutes

Calculate the areas of the following road and town developments, assuming that the measurements are sufficiently accurate to allow answers to the nearest whole number. Remember to click on ‘reveal comment’ for a hint or tip.

  • a.A roundabout of radius 15.6 m.

Answer

  • a.Area of a circle equals pi multiplication radius squared

    multiline equation line 1 So the area of the roundabout equation left hand side equals right hand side pi postfix multiplication left parenthesis 15.6 m right parenthesis squared line 2 equals 765 m super two

  • b.A circular building with a diameter of 54 metres.

Comment

Remember that you need the radius, not the diameter for the formula for area of a circle.

Answer

  • b.Here you are given the diameter. The radius is half the diameter.

    multiline equation line 1 So radius equals 54 m prefix division of two line 2 equals 27 m

    multiline equation line 1 So comma the area of the circular building equation left hand side equals right hand side pi postfix multiplication left parenthesis 27 m right parenthesis squared line 2 equals 2290 m super two left parenthesis to the nearest whole number right parenthesis

  • c.A semicircular lecture hall with a diameter of 46 m.

Answer

  • c.The area of a semicircle is half the area of a circle.

    So the area of a semicircle equation left hand side equals right hand side one divided by two multiplication pi multiplication radius squared

    The lecture hall has a diameter of 46 m so the radius is 23 m.

    multiline equation line 1 The area of the lecture hall equation left hand side equals right hand side one divided by two multiplication pi postfix multiplication left parenthesis 23 m right parenthesis squared line 2 equals 831 m super two left parenthesis to the nearest whole number right parenthesis full stop

The last few sections have included a number of different formulas, so you might appreciate a quick summary to bring them altogether. You might also like to take a note of them alongside any new vocabulary you’ve come across

  • area of a rectangle equals length multiplication width
  • area of a parallelogram equals base multiplication height
  • area of a triangle equation left hand side equals right hand side one divided by two postfix multiplication base multiplication height
  • area of a circle equation left hand side equals right hand side pi multiplication radius super two full stop

From your study of measurement in Week’s 1 and 2 of the course you will know that there is one final property of shapes that hasn’t been covered, that is capacity or volume. So, you’ll move onto that now in the final part of this week’s study.

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