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Surface areas of advanced solids and composites

Optional extension for this course. It is not required for core topic progress.

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A sphere’s radius increases from 2 cm to 4 cm. What happens to its surface area?

Year 10, Chapter 1, 4 difficulty levels, Calculator allowedBuild a worksheet

Count exposed surfaces

Name the surfaces before choosing a formula. Use one unit throughout and round only the final result

\[\begin{gathered}A_{\text{sphere}}=4\pi r^2\\A_{\text{cone wall}}=\pi rs\\A_{\text{hemisphere wall}}=2\pi r^2\end{gathered}\]
  1. Open or closed Add the circular base πr² only when it is included
  2. Slant and perpendicular heights Cone area uses s = √(r² + h²). Volume uses perpendicular height h.
  3. Joined solids Add only exposed surfaces, or add complete surface areas and subtract the contact area twice
  4. Volume Add non-overlapping component volumes. Contact faces are boundaries, not lost volume

Worked example

A cylinder of radius 2 cm and height 6 cm is topped with one hemisphere of the same radius. Their circular faces touch fully. Find all exposed area, including the bottom, to 2 decimal places

\[A=A_w+A_d+A_b\]
  1. \(\displaystyle A_{\text{wall}}=2\pi(2)(6)=24\pi\)Cylinder curved wall, in cm²
  2. \(\displaystyle A_{\text{dome}}=2\pi(2)^2=8\pi\)Only the hemisphere’s curved surface, in cm²
  3. \(\displaystyle A_{\text{bottom}}=\pi(2)^2=4\pi\)The bottom circle is exposed, in cm²
  4. \(\displaystyle A=36\pi\approx113.10\text{ cm}^2\)Round the final area to 2 decimal places

Common mistakes

Using cone height for surface area
✕\(\displaystyle A=\pi rh\)✓\(\displaystyle A=\pi rs\)
s is the slant height measured along the wall
Forgetting a closed hemisphere’s base
✕\(\displaystyle A=2\pi r^2\)✓\(\displaystyle A=3\pi r^2\)
The curved surface is 2πr² and the included circular base contributes πr²
Counting a joined face once
✕\(\displaystyle A=A_1+A_2-C\)✓\(\displaystyle A=A_1+A_2-2C\)
The separate complete areas counted two copies of the matching contact face

Difficulty levels

introSphere surface area from radius or diameter
standardOpen and closed cones and hemispheres
challengeRectangular pyramids and joined solids; compare exposed area with volume
examRecognise composite surface areas and calculate their volumes

Questions about this topic

Can a sphere be unfolded into a flat net?+

Not without distortion. The central circle is a cross-section, and its area is one quarter of the sphere surface area

Why does a cone use two different heights?+

Perpendicular height measures distance to the base plane. Slant height lies on the wall, so it becomes the radius of the unfolded sector

Do all pyramid faces have the same slant height?+

A right square-based pyramid does. A right rectangular pyramid generally has two pairs of different slant heights

Is subtracting the contact face twice always valid?+

It applies when adding complete surface areas of two solids joined along matching faces. If you instead list only exposed faces, do not subtract again

Does coating area equal the amount of material to purchase?+

Only in an ideal model without waste, overlap or thickness. Real purchasing decisions may need allowances, coverage rates and further measurements

In this chapter

Convert measurement unitsPerimeter of standard shapesFind an unknown side from a perimeterCircumference of a circlePerimeter of composite shapesArea of standard shapesArea of a circleArea of a sectorArea of composite shapes1F Prism surface area: Surface area of solidsSurface area of a triangular prismExposed surface area of joined cuboids1G Cylinder surface area: Surface area of solidsSurface area of a cylinder portionSurface area of a cylinder joined to a cuboidVolume of prisms and cylindersCapacity of a containerVolume of a cylinder portionJoined and hollow prism-cylinder volumesModel rainwater collection and storageSmallest unit and limits of accuracyMeasurement bounds in calculationsVolume of cones, pyramids and spheres (optional extension)1J Level 10A extension: pyramids, cones and spheres: Surface area of solids (optional extension)Surface areas of advanced solids and composites (optional extension)