Year 10 Standard · Unit 5 Solids, similarity and transformations · Lesson 3 of 6

How much skin does a sphere have, and which faces does a joined solid keep?

A sphere's surface can be compared with a circle of the same radius. Joining solids hides some faces, so the surface of a composite solid is only the outside faces added up.

Geometry and TrigonometryGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: a cylinder has radius 4 cm and height 6 cm. Find its curved surface area in terms of π.
    Answer48π cm², because 2π × 4 × 6 = 48π.
  2. From the last lesson: find the volume of a sphere of radius 3 cm in terms of π.
    Answer36π cm³, because 4π × 27 ÷ 3 = 36π.
  3. From Year 9: a circle has diameter 10 cm. Find its area in terms of π.
    Answer25π cm², because the radius is 5 and π × 5² = 25π.
Words used in this lesson
sphere
a perfectly round solid, every point the same distance from the centre. For example: a ball.
hemisphere
half of a sphere. For example: half a ball.
great circle
the biggest circle you can draw on a sphere, through its centre. For example: the equator.
composite solid
a solid made by joining simpler solids. For example: a toy with a cone fixed to a cylinder.
outside surface
the faces you can touch from outside the solid. For example: base, side and top.
dome
the curved top of a hemisphere. For example: the top of a mosque.

The peel of an orange

In a market in Marrakesh a juice seller peels an orange in one piece. She wonders how many flat circles, each as wide as the orange across its middle, she could cut out of the peel if she pressed it flat.

Predict first: would the peel cover 2 of those circles, 3, or 4? Write down a reason, then test it: draw round an orange on paper, peel it, press the peel flat and see how many of those circles it fills. Then use the first figure.

2
12.57one circle, πr²
50.27surface area of the sphere

Radius 2: one circle is πr² = 4π ≈ 12.57. The sphere's surface is 4πr² = 16π ≈ 50.27, which is 4.00 times one circle.

Show me the steps
  1. Radius 2: one circle is π × 2² = 4π ≈ 12.57. The sphere's surface is 4π × 2² = 16π ≈ 50.27.
  2. Radius 1: one circle is π ≈ 3.14 and the sphere is 4π ≈ 12.57.
  3. Radius 5: one circle is 25π ≈ 78.54 and the sphere is 100π ≈ 314.16.
  4. In every state the ratio of the sphere's surface to one circle of the same radius is 4.00, so the surface area is 4πr².

The surface area of a sphere is 4πr², four times the area of its great circle. A solid hemisphere has a curved surface 2πr² and a flat circular face πr², so its total is 3πr².

A composite solid has only its outside faces counted. When two solids are joined, the faces that touch are hidden inside, so they are not part of the surface. The volume is the sum of the parts.

2
3
3
75.40total outside surface area
54.45volume

Radius 2, cylinder height 3, hemisphere on top: base 12.57, side 37.70, top 25.13, so the outside surface is 75.40. Volume ≈ 54.45.

Show me the steps
  1. Radius 2, cylinder height 3, hemisphere on top: base π × 2² = 4π, side 2π × 2 × 3 = 12π, dome 2π × 2² = 8π. The outside is 24π ≈ 75.40.
  2. Press Cone with a cone height of 3: the cone's slant height is √(2² + 3²) = √13, so the top is π × 2 × √13 ≈ 22.65 and the outside surface is 16π + 22.65 ≈ 72.92.
  3. Make the cone height 6: s = √40 ≈ 6.32, the top is ≈ 39.74, and the volume is 12π + 8π = 20π ≈ 62.83. A taller spike adds area and volume.
  4. In every state the circle where the top joins the cylinder is missing from the total: it is inside the solid.

Count the faces you can touch. A cylinder with a dome on top has two parts but only three outside surfaces: the base, the curved side and the dome. Adding the circle where they meet counts a face that is inside the solid.

Worked examples

1. A sphere has radius 15 cm. Find its surface area in terms of π and to 1 decimal place.

  1. Surface area = 4π × 15² = 4π × 225 = 900π.
  2. 900π ≈ 2827.4 cm².

2. A solid hemisphere has radius 10 cm. Find its total surface area.

  1. Curved surface = 2π × 10² = 200π. Flat face = π × 10² = 100π.
  2. Total = 300π ≈ 942.5 cm².

3. A cylinder of radius 5 cm and height 14 cm has a hemisphere of radius 5 cm on top. Find the outside surface area.

  1. Base = π × 5² = 25π. Side = 2π × 5 × 14 = 140π. Dome = 2π × 5² = 50π.
  2. Total = 25π + 140π + 50π = 215π ≈ 675.4 cm².

Your go. The first step is given. A capsule is a cylinder of radius 2 cm and height 7 cm with a hemisphere of radius 2 cm fixed to each end. Find its surface area in terms of π.

  1. The two hemispheres together make one whole sphere of radius 2, with surface 4π × 2².
  2. Now add the curved side of the cylinder: 2π × 2 × 7.
Check the answer

Sphere = 16π, side = 28π, total = 44π cm² (≈ 138.2 cm²). There are no flat faces: both ends are joined.

With a GDC (optional)

Enter the whole expression once and round only at the end: 4 × π × 11². Check the exact form first, such as 484π, so that you can see whether the decimal is sensible.

A GDC does not decide which faces are outside. Draw the solid and list the faces first. Everything here can be done by hand with π left in the answer.

Check before you go on

1. A solid hemisphere has a curved surface area of 50π cm². What is its total surface area, including its flat face?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A ball has a diameter of 16 cm. Its surface area is ?π cm². What number goes in front of π?

3. A solid hemisphere has radius 7 cm. Its total surface area, including the flat face, is ?π cm². What number goes in front of π?

Level 2 · Building

4. A football has radius 11 cm. What is its surface area, in cm², to the nearest whole number?

5. A sphere has a surface area of 324π cm². What is its radius, in cm?

6. A solid is a cylinder of radius 6 cm and height 5 cm with a hemisphere of radius 6 cm on top. Its outside surface area is ?π cm². What number goes in front of π?

Level 3 · Stretch

7. For the same solid, a cylinder of radius 6 cm and height 5 cm with a hemisphere of radius 6 cm on top, the volume is ?π cm³. What number goes in front of π?

8. A toy is a cylinder of radius 4 cm and height 10 cm with a cone of radius 4 cm and vertical height 3 cm fixed on top. Its outside surface area is ?π cm². What number goes in front of π?

Something to think about

A sphere of radius r fits snugly inside a cylinder of height 2r. In lesson 2 the sphere filled two thirds of the cylinder's volume. Does it also fill two thirds of the cylinder's surface? Work out the cylinder's total surface area, with its lids, in terms of r.

One way to think about it

The cylinder's total surface is 2πr² + 2πr × 2r = 6πr², and the sphere's is 4πr². So the sphere has 4/6 = two thirds of the total surface too.

Exit ticket

9. A ball of radius 6 cm just fits inside a cylindrical tube of radius 6 cm and height 12 cm, with a lid at each end. Which statement about surface areas is true?