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Themed maths

Spring egg geometry for IGCSE and GCSE

A ready-to-teach lesson for the spring term: model an egg with a hemisphere and a cone, with an ellipse, and with similar solids. A 5-minute starter, a 35-minute main activity, an extension and full worked answers.

Level
IGCSE and GCSE (Higher and Extended)
Time
40 minutes, plus a 10-minute extension
Topics
Volume and surface area of spheres, hemispheres and cones; Composite solids; Similar shapes: area and volume scale factors; Area of an ellipse (given formula)
Equipment
The starter is non-calculator. A calculator is needed for the main activity.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A12 minA hemisphere-and-cone egg
Main: task B11 minEllipse eggs
Main: task C12 minSimilar chocolate eggs
Extension10 minFast finishers or homework

Starter (5 minutes)

No calculator. Leave π in your answers.

  1. Find the volume of a sphere of radius 3 cm.
  2. Find the volume of a hemisphere of radius 3 cm.
  3. Find the volume of a cone with radius 3 cm and height 4 cm.
  4. A shape is enlarged by scale factor 2. What happens to its volume?

Main activity (35 minutes)

Task A: A hemisphere-and-cone egg (12 min)

A model egg is a hemisphere of radius 3 cm joined to a cone of radius 3 cm and height 5 cm.

  1. Find the volume of the egg, exactly and to 3 significant figures.
  2. Find the surface area of the egg, to 3 significant figures.
  3. A box holds 6 of these eggs made of solid chocolate. Find the total volume of chocolate, to 3 significant figures.

Task B: Ellipse eggs (11 min)

An egg’s outline is an ellipse with half-length a and half-width b. Its area is πab, and the egg-shaped solid has volume 4⁄3πab2.

  1. Take a = 3 cm and b = 2 cm. Find the area of the outline, to 3 significant figures.
  2. The egg is enlarged so that a = 4.5 cm and b = 3 cm. Find the new area using the area scale factor, to 3 significant figures.
  3. Find the volume of the solid egg with a = 3 cm and b = 2 cm, to 3 significant figures.

Task C: Similar chocolate eggs (12 min)

Two chocolate eggs are mathematically similar. The small one is 6 cm tall and the large one is 9 cm tall.

  1. Find the length scale factor and the volume scale factor.
  2. The small egg contains 40 g of chocolate. How much chocolate is in the large egg?
  3. The large egg’s foil wrapper has area 90 cm2. What is the area of the small egg’s wrapper?

Extension (10 minutes)

For fast finishers, or as homework.

  1. An egg is a hemisphere of radius r on a cone of radius r and height 5r/3. Its volume is 100 cm3. Find r, to 3 significant figures.
  2. A large chocolate egg has 8 times the volume of a small similar egg. The small egg is 5 cm tall. How tall is the large one?

For teachers

Teacher notes and full worked answers
  • Bring in a real egg and a ruler: students can compare their measurements with the models.
  • In task A (b), the flat circle where the hemisphere meets the cone is inside the egg, so it is not part of the surface area.
  • In task C, the common slip is to use the length scale factor for volume or area.

Starter

  1. 36π cm3
    • 4⁄3π × 27 = 36π
  2. 18π cm3
    • Half of 36π
  3. 12π cm3
    • 1⁄3π × 9 × 4 = 12π
  4. It is multiplied by 8
    • 23 = 8

Task A: A hemisphere-and-cone egg

  1. 33π ≈ 104 cm3
    • Hemisphere: 18π. Cone: 1⁄3π × 9 × 5 = 15π.
    • 33π = 103.67…
  2. 112 cm2
    • Curved hemisphere: 2π × 9 = 18π.
    • Slant height l = √(9 + 25) = √34; curved cone: π × 3 × √34.
    • 18π + 3π√34 = 56.55… + 54.95… = 111.5…
  3. 622 cm3
    • 6 × 33π = 198π = 622.0…

Task B: Ellipse eggs

  1. 18.8 cm2
    • π × 3 × 2 = 6π = 18.84…
  2. 42.4 cm2
    • Scale factor 1.5, so area scale factor 2.25.
    • 6π × 2.25 = 13.5π = 42.41…
  3. 50.3 cm3
    • 4⁄3π × 3 × 4 = 16π = 50.26…

Task C: Similar chocolate eggs

  1. 1.5 and 3.375
    • 9 ÷ 6 = 1.5; 1.53 = 3.375
  2. 135 g
    • 40 × 3.375 = 135
  3. 40 cm2
    • Area scale factor 1.52 = 2.25. 90 ÷ 2.25 = 40

Extension

  1. 2.96 cm
    • Volume = 2⁄3πr3 + 1⁄3πr2 × 5r/3 = (2/3 + 5/9)πr3 = (11/9)πr3.
    • r3 = 900/(11π) = 26.04…, so r = 2.964…
  2. 10 cm
    • Length scale factor = ∛8 = 2, so 5 × 2 = 10 cm.

The same answers are in the answers PDF. Every answer was recomputed by computer before publishing.

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