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      Lesson details

      Learning outcome

      I can calculate the surface area of a cone.

      Key learning points

      1. The slanted surface of a cone can be cut into congruent triangles.
      2. These form a parallelogram with dimensions of the slanted length and half of the circumference of the base.
      3. The area of the slanted surface can be written as πrl
      4. The surface area of the cone is therefore πrl + πr^2
      5. The slanted length can be calculated from the height of the cone and radius of the base

      Keywords

      • Cone - A circular cone is a 3D shape that has a circular base and one curved surface that narrows from the base to a fixed point called the apex/vertex.

      • Right cone - A line drawn through the apex and centre of the base of a right circular cone will be perpendicular to the base.

      • Apex - The apex is the point (vertex) which is the greatest perpendicular distance from the base.

      Common misconception

      When finding the surface area of the cone, pupils may find the curved surface area but forget to include circular base of the cone.

      By re-reading the question, and inspecting any given diagram, pupils can check if the cone is open or closed and therefore if they need to include the circular base in their calculation.

      Teacher tip

      Encourage pupils to clearly label on diagrams and in their solutions, if they have been given the radius or diameter of the circular base of the cone.

      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.
      Match each measurement to the correct formula.

      Correct Answer:Surface area of a sphere,$$4 \pi r^2$$

      $$4 \pi r^2$$

      Correct Answer:Volume of a sphere,$$\frac{4}{3} \pi r^3$$

      $$\frac{4}{3} \pi r^3$$

      Correct Answer:Surface area of a cylinder,$$2 \pi rh + 2 \pi r^2$$

      $$2 \pi rh + 2 \pi r^2$$

      Correct Answer:Volume of a cylinder,$$ \pi r^2 h$$

      $$ \pi r^2 h$$

      Q2.
      Work out the volume of a sphere with radius 1.4 m. Give your answer correct to 2 decimal places.

      8.21 m³
      Correct answer: 11.49 m³
      25.86 m³
      34.48 m³

      Q3.
      A sphere has a radius of 9 mm. The volume of the sphere, in terms of 𝜋, is 𝜋 mm³.

      Correct Answer: 972

      Q4.
      Which of these calculations finds the volume of a hemisphere with a radius of 6 cm?

      Correct answer: $$ \frac{4}{3} \times \pi \times 6^3 \div 2$$
      Correct answer: $$ \frac{1}{2} \times \frac{4}{3} \times \pi \times 6^3$$
      $$ \frac{4}{3} \times \pi \times 3^3 \div 2$$
      $$ \frac{2}{6} \times \pi \times 6^3 $$
      Correct answer: $$ \frac{2}{3} \times \pi \times 6^3 $$

      Q5.
      The volume of a hemisphere with a radius of 6 cm, in terms of $$\pi$$, is $$\pi$$ cm³.

      Correct Answer: 144

      Q6.
      The volume of a hemisphere is 1152𝜋 cm³. The surface area of the hemisphere, in terms of 𝜋, is 𝜋 cm².

      An image in a quiz
      Correct Answer: 432

      6 Questions

      Q1.
      Which of these diagrams is a cone?

      An image in a quiz
      Correct Answer: An image in a quiz
      An image in a quiz
      An image in a quiz
      An image in a quiz

      Q2.
      The curved surface of this cone is unfolded into a circular sector. Match each measurement to its value.

      An image in a quiz
      Correct Answer:The radius of the sector is…,10 cm

      10 cm

      Correct Answer:The arc length of the sector is…,8𝜋 cm

      8𝜋 cm

      Correct Answer:The area of the sector is…,40𝜋 cm²

      40𝜋 cm²

      Q3.
      Calculate the area of the curved surface of this cone. Give your answer correct to 3 significant figures.

      An image in a quiz
      84.0 cm²
      Correct answer: 132 cm²
      160 cm²
      264 cm²

      Q4.
      Calculate the total surface area of this cone. Give your answer correct to 3 significant figures.

      An image in a quiz
      78.5 cm²
      188 cm²
      267 cm²
      Correct answer: 283 cm²

      Q5.
      The slant length of this cone is cm. Give your answer correct to 2 decimal places.

      An image in a quiz
      Correct Answer: 16.97

      Q6.
      The total surface area of this cone is cm². Give your answer correct to 3 significant figures.

      An image in a quiz
      Correct Answer: 1090, 1,090, 1 090

      To help you plan your 11 maths lesson on: The surface area of a cone, download all teaching resources for free and adapt to suit your pupils' needs...