Arc-Lengths-And-Sector-Areas Aqa Foundation

Arc lengths & sector areas

What is an arc?

• An arc is a part of the circumference of a circle 

• Two points on a circumference of a circle will create two arcs 

  • The smaller arc is known as the minor arc

  • The bigger arc is known as the major arc

What is a sector?

• A sector is the part of a circle enclosed by two radii (radiuses) and an arc

  • A sector looks like a slice of a circular pizza

  • The curved edge of a sector is the arc

• Two radii in a circle will create two sectors

  • The smaller sector is known as the minor sector

  • The bigger sector is known as the major sector

What formulae do I need to know?

• You need to be able to calculate the length of an arc and the area of a sector

• The angle formed in a sector by the two radii is often labelled θ (the Greek letter “theta”)

• You can calculate the area of a sector or the length of an arc by adapting the formulae for the area or circumference of a circle

  • A full circle is equal to 360° so the fraction will be the angle, θ°, out of 360°

    • 

    • 

• Working with sector and arc formulae is just like working with any other formula:

  • Write down what you know (or what you want to know)

  • Pick the correct formula

  • Substitute the values in and solve

How do I find the length of an arc?

• STEP 1
  Divide the angle by 360 to form a fraction

  • 

• STEP 2
  Calculate the circumference of the full circle

  • 

• STEP 3
  Multiply the fraction by the circumference

  • <img alt=”theta over 360 cross times 2 straight pi r” data-mathml='<math ><semantics><mrow><mfrac><mi>&#952;</mi><mn>360</mn></mfrac><mo>&#215;</mo><mn>2</mn><mi mathvariant=”normal”>&#960;</mi><mi>r</mi></mrow><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″,”autoformat”:true}</annotation></semantics></math>’ height=”47″ role=”math” src=”data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2247%22%20width%3D%2281%22%20wrs%3Abaseline%3D%2230%22%3E%3C!–MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfrac%3E%3Cmi%3E%26%23x3B8%3B%3C%2Fmi%3E%3Cmn%3E360%3C%2Fmn%3E%3C%2Fmfrac%3E%3Cmo%3E%26%23xD7%3B%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmi%20mathvariant%3D%22normal%22%3E%26%23x3C0%3B%3C%2Fmi%3E%3Cmi%3Er%3C%2Fmi%3E%3C%2Fmath%3E–%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A’math10cd7b9978ec9654b7f29750478’%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIA