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      Area of composite shapes

      Lesson details

      Learning outcome

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

      Key learning points

      1. The area parts of a circle can be found using the area formula and reasoning.
      2. The area of shapes made from circles and parts of circles can be found.
      3. The area of composite shapes that include circles can be found.

      Keywords

      • Sector - A sector is the region formed between two radii and their connecting arc.

      Common misconception

      To find the area of a quarter-circle divide the radius by 4 to use as part of the calculation.

      You must divide the area by 4, not the radius. The radius gets squared when calculating area so the division of r would be applied twice.

      Teacher tip

      Encourage students to sketch the component parts of each composite shape. This is especially helpful in visualising the shape of both the negative and positive space in a composite shape with a component part removed from the object.

      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 $${2\over3}$$ of 144 cm?

      96
      Correct answer: 96 cm
      216
      216 cm

      Q2.
      Which of these formulae finds the area of a circle with radius $$r$$?

      $$area = {\pi} {\times}r$$
      $$area = 2{\times}{\pi} {\times}r$$
      $$area = {\pi} {\times}r2$$
      Correct answer: $$area = {\pi} {\times}r^2$$

      Q3.
      Find the area of a circle with radius of 25 cm, giving your answer both in terms of $$\pi$$ and in decimal form, rounded to 1 decimal place.

      25$$\pi$$ cm²
      50$$\pi$$ cm²
      Correct answer: 625$$\pi$$ cm²
      1963.4 cm²
      Correct answer: 1963.5 cm²

      Q4.
      From its centre, this circle has been split up into 9 equally-sized sectors. How many degrees is the angle marked x?

      An image in a quiz
      Correct Answer: 40, 40°, 40degrees, 40 degrees

      Q5.
      Match each statement to the correct value for a circle with a diameter of 80 units.

      Correct Answer:The radius is,40
      Correct Answer:The area in terms of $$\pi$$,1600$$\pi$$
      Correct Answer:The area in decimal form is,5026.5
      Correct Answer:The circumference is,80$$\pi$$

      Q6.
      Which of these statements is true for this composite shape?

      An image in a quiz
      $$a$$ = 7 cm
      Correct answer: $$a$$ = 15 cm
      $$a$$ = 18 cm
      Correct answer: The area of the composite shape is 1128 cm².
      The area of the composite shape is 1190 cm².

      6 Questions

      Q1.
      A quarter-circular sector is cut out of a circle. The original circle had an area of 120$$\pi$$ cm². What is the area of the quarter-circle?

      Correct answer: 30$$\pi$$ cm²
      40$$\pi$$ cm²
      94.25$$\pi$$ cm²
      480$$\pi$$ cm²
      Correct answer: 94.25 cm²

      Q2.
      A sector with a 60° angle is cut out of a circle. The original circle had an area of 3072 inches². The area of the sector is square inches.

      Correct Answer: 512, 512 square inches, 512 inches²

      Q3.
      Which of the following statements is true for this sector?

      An image in a quiz
      This sector is $$2\over3$$ the area of a circle that has the same radius.
      Correct answer: This sector is $$3\over4$$ the area of a circle that has the same radius.
      The area of this sector is 16.5$$\pi$$ cm².
      Correct answer: The area of this sector is 363$$\pi$$ cm².
      The area of this sector is 484$$\pi$$ cm².

      Q4.
      This composite shape is broken down into its component parts, made from circular sectors and quadrilaterals. Match each component part and composite shape to its area.

      An image in a quiz
      Correct Answer:area of square,4 cm²
      Correct Answer:area of semicircle,$${1\over2}\pi$$ cm²
      Correct Answer:area of quarter-circle,3.14 cm²
      Correct Answer:area of composite shape,8.71 cm²

      Q5.
      Which of these shows the area (written in terms of $$\pi$$) of the shaded region of this composite shape composed of a circular sector and a square?

      An image in a quiz
      2304 cm²
      (2304 + 576$$\pi$$) cm²
      (2304 + 288$$\pi$$) cm²
      (2304 − 576$$\pi$$) cm²
      Correct answer: (2304 − 288$$\pi$$) cm²

      Q6.
      What is the area (in cm²) of this composite shape, composed of only squares and circular sectors, rounded to the nearest integer? cm²

      An image in a quiz
      Correct Answer: 393, 393 cm², 393cm²

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