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      Perimeter of composite shapes

      Lesson details

      Learning outcome

      I can solve perimeter problems of composite shapes involving whole and/or part circles.

      Key learning points

      1. The length of fractions of the circumference of a circle can be found.
      2. The perimeter of parts of a circle can be found using the formula for the circumference and reasoning.
      3. The perimeter of shapes made from circles and parts of circles can be found using the circumference and reasoning.
      4. The perimeter of composite shapes including circles can be found.

      Keywords

      • Arc - An arc is part of a curve. When describing an arc of a circle, it is part of the circle's circumference.

      Common misconception

      Shapes with the same area also have the same perimeter.

      Demonstrate that component parts being rearranged has no effect on its area, but may affect its perimeter.

      Teacher tip

      Doubling the angle of the sector does not double its perimeter. For example, a semicircle does not have double the perimeter of a quarter-circle with the same radius, as the lengths of the two radii bounding the arc have not doubled.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      What is $${1\over4}$$ of 76$$\pi$$ cm? Give your answer in decimal form and rounded to 2 decimal places.

      Correct Answer: 59.69 cm, 59.69cm

      Q2.
      Which of these units of measurement is appropriate for measuring the perimeter of something?

      Correct answer: inches
      inches²
      Correct answer: mm
      mm³
      degrees

      Q3.
      Which of these formulae finds the circumference of a circle with radius $$r$$ and diameter $$d$$?

      $$circumference = {\pi} {\times}r$$
      Correct answer: $$circumference = 2{\times} {\pi} {\times}r$$
      Correct answer: $$circumference = {\pi} {\times}d$$
      $$circumference = {\pi} {\times}d^2$$
      $$circumference = 2{\times}{\pi} {\times}d$$

      Q4.
      Which of these describes the circumference of a circle with a diameter of 60 cm?

      30$$\pi$$ cm
      Correct answer: 60$$\pi$$ cm
      120$$\pi$$ cm
      94.2 cm
      Correct answer: 188.5 cm

      Q5.
      A circular hoop with a radius of 40 cm is cut into 12 equally-sized pieces. What is the length of one piece? Give your answer in cm, rounded to 1 decimal place. cm

      Correct Answer: 20.9, 20.9 cm, 20.9cm

      Q6.
      Circle A has a radius of 49 cm. Circle B has a diameter of 1.33 metres. How much longer is the circumference of circle B than circle A? Give your answer in cm, rounded to the nearest integer.

      Correct Answer: 110, 110 cm, 110cm

      6 Questions

      Q1.
      A quarter-circular sector is cut out of a circle. The original circle had a perimeter of 60$$\pi$$ cm. What is the arc length of this sector?

      Correct answer: 15$$\pi$$ cm
      15$$\pi$$ + 30 cm
      15$$\pi$$ + 60 cm

      Q2.
      A semicircular sector is cut out of a circle. The radius of the original circle is 31 cm, and its circumference is 194.8 cm (rounded to 1 decimal place). What is the perimeter of the sector?

      31$$\pi$$
      97.4 cm
      97.4 + 31 cm
      Correct answer: 97.4 + 62 cm
      Correct answer: 159.4 cm

      Q3.
      Calculate the perimeter of this sector, giving your answer in cm and rounding your answer to 1 decimal place.

      An image in a quiz
      Correct Answer: 294.8, 294.8 cm, 294.8cm

      Q4.
      This composite shape is composed of a circular sector and a square. Which of these statements are true about the perimeter of this composite shape that has been broken into component lengths.

      An image in a quiz
      There are three line segments of 68 cm.
      Correct answer: There are four line segments of 68 cm.
      There are five line segments of 68 cm.
      The length of the arc is 68$$\pi$$ cm.
      Correct answer: The length of the arc is 34$$\pi$$ cm.

      Q5.
      This composite shape is composed of two quarter-circles and a rectangle. Match the statement about the component parts of the composite shape's perimeter to the correct value.

      An image in a quiz
      Correct Answer:value of $$a$$ ,9 cm

      9 cm

      Correct Answer:radius of each quarter-circle,8 cm

      8 cm

      Correct Answer:total length of line segments,34 cm

      34 cm

      Correct Answer:total length both arcs,8$$\pi$$ cm

      8$$\pi$$ cm

      Correct Answer:total perimeter of shape,59.1 cm

      59.1 cm

      Q6.
      The square at the centre of this composite shape has an area of 441 cm². Find the perimeter of the composite shape in cm, rounded to the nearest integer.

      An image in a quiz
      Correct Answer: 216, 216 cm, 216cm

      To help you plan your 8 maths lesson on: Perimeter of composite shapes, download all teaching resources for free and adapt to suit your pupils' needs...