Mathematics Circles

Lesson Objectives The aim of this powerpoint is to help you… to review the terminology linked to different parts of a circle to learn and use the formula for calculating the circumference of a circle. to learn and use the formula for calculating the area of a circle.

Terminology Circumference is the distance all the way around the circle Arc is part of the circumference Diameter is the distance from one side of a circle to the other through the centre Radius is the distance from the centre to the circumference (half of the diameter)

Terminology Chord is the line which joins one side of the circle with another NOT through the centre and splits the circle into two pieces called segments The smaller segment is called the minor segment and the larger one is called the major segment Two radii cut the circle into pieces called sectors

Pi (symbol: π) Pi is an irrational number – it can not be expressed as a proper fraction. This means that it has a decimal value that never stops and the decimals do NOT have a repeating pattern. Pi is a constant (like any number by itself) Pi is worth… 3.1416927………….. Some calculators have a pi button otherwise we usually round pi’s value to 3.14 or 31/7 (or 22/7)

Finding the Circumference The formula for calculating the length of a circle’s circumference is: C = πd (ie. pi × diameter) But remember that diameter = 2 × radius so… C = 2πr (ie. pi × 2 × radius)

Using the Formulae A circle has a radius of 5cm. What is its (a) diameter & (b) circumference? (a) d = 2 × radius = 2 × 5cm = 10cm (b) C = pi × diameter = 3.14 × 10cm = 31.4cm A circle has a diameter of 12m. What is its circumference? (c) C = pi × diameter = 3.14 × 12m = 37.68m

What next? Print out the notes called PAV3-Circles. Read through them and make sure you answer any questions on the parts of a circle and finding the circumference of a circle. Work through the MyMaths lesson and its online homework called Circumference of a Circle found at: http://app.mymaths.co.uk/269-resource/circumference-of-a-circle http://app.mymaths.co.uk/269-homework/circumference-of-a-circle Now continue working through this powerpoint

Estimating the Area of a Circle Look at the square 1 unit is the distance from the edge to the centre. It has an area of 4 square units. Look at each individual section and putting them altogether, the circle’s area is just over ¾ of the grid’s area so it has an area of just over 3 square units. SUMMARY: If radius is 1, then radius × radius is 1 square unit. This circle’s area is just over 3 (ie. pi) of these units.

Finding the Area The formula for calculating the area of a circle is: A = πr² (ie. pi × radius × radius) Remember that if you are given the length of the diamater you will need to use it to calculate the length of the radius first which is: radius = diameter ÷ 2

Using the Formulae A circle has a radius of 5cm. What is its area? (a) A = pi × radius × radius = 3.14 × 5cm × 5cm = 78.5cm² A circle has a diameter of 12m. What is its (b) radius & (c) area? (b) r = ½ of diameter = 12m ÷ 2 = 6m (c) A = pi × radius × radius = 3.14 × 6m × 6m = 113.04m²

Circular Shapes & Common Sense Find to 2 d.p. the perimeter and area of the semicircle shown below: 0.3m d = 0.3m so r = 0.3 ÷ 2 = 0.15m Perimeter will be the diameter along the bottom plus the circumference of HALF a circle C = 3.14 × 0.3m = 0.942m ½ of C = 0.942 ÷ 2 = 0.471m P = 0.3m + 0.47m = 0.77m (to 2 d.p.) Area is for HALF a circle! A = 3.14 × 0.15m × 0.15m = 0.0707m² ½ of A = 0.0707 ÷ 2 = 0.04m² (to 2 d.p.)

Working Backwards You may be given the area or circumference and have to work backwards to find the radius or diameter. You will need to divide or take square roots! A circle has an area of 50.27mm². Find its circumference. To find the circumference we must first find the radius working backwards using the area! A = πr² so 50.27 = 3.14 × r² Divide by 3.14: 16 = r² Take square roots: 4 = r Now use C = 2πr  2 × 3.14 × 4 = 25.12mm

What next? Finish reading through the notes, making sure you answer any further questions. Work through the MyMaths lesson and its online homework called Area of a Circle found at: http://app.mymaths.co.uk/270-resource/area-of-a-circle http://app.mymaths.co.uk/270-homework/area-of-a-circle Save and complete the worksheet called Circles-S1.xlsx Now move on to the PAV4 powerpoint