Surface area (Edexcel GCSE Maths): Revision Notes

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Surface area

Surface area is the total area of all the outer surfaces of a 3D shape. Understanding how to calculate surface area is essential for solving geometry problems and real-world applications.

Cone surface area

A cone has two types of surface area you need to understand:

Curved surface area

The curved surface area only includes the slanted side of the cone, not the circular base.

Formula: Curved surface area of cone = πrl\pi rl

infoNote

In this formula, ll represents the slant height (the distance from the apex to the edge of the base), not the vertical height of the cone.

Where:

  • rr = radius of the base
  • ll = slant height (the distance from the apex to the edge of the base)

Total surface area

The total surface area includes both the curved surface and the circular base.

Formula: Total surface area of cone = πr2+πrl\pi r^2 + \pi rl

This combines:

  • Base area = πr2\pi r^2
  • Curved surface area = πrl\pi rl

Finding the slant height

When you're given the vertical height and radius, use Pythagoras' theorem to find the slant height:

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Worked Example: Finding Slant Height

Given: A cone with radius r=3r = 3 cm and vertical height h=4h = 4 cm

Step 1: Draw a right-angled triangle using the radius, vertical height, and slant height

Step 2: Apply Pythagoras' theorem l2=h2+r2l^2 = h^2 + r^2

Step 3: Substitute the values l2=42+32=16+9=25l^2 = 4^2 + 3^2 = 16 + 9 = 25

Step 4: Find the slant height l=25=5l = \sqrt{25} = 5 cm

  • Draw a right-angled triangle using the radius, vertical height, and slant height
  • Use l2=h2+r2l^2 = h^2 + r^2 to calculate the slant height

General approach for cone problems

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Worked Example Approach

  1. Identify what measurements you have (radius, height, slant height)
  2. Calculate any missing measurements using Pythagoras
  3. Substitute values into the appropriate formula
  4. Calculate your final answer

Sphere surface area

The surface area of a sphere is much simpler to calculate than a cone.

Formula: Surface area of sphere = 4πr24\pi r^2

Where rr is the radius of the sphere.

infoNote

The sphere formula can be remembered as four times the area of a great circle (πr2\pi r^2). A great circle is any circle that passes through the centre of the sphere.

Hemisphere surface area

A hemisphere is exactly half a sphere, so its curved surface area is:

Curved surface area of hemisphere = 12×4πr2=2πr2\frac{1}{2} \times 4\pi r^2 = 2\pi r^2

chatImportant

Remember that if you need the total surface area of a hemisphere, you must also add the area of the flat circular base (πr2\pi r^2).

Total surface area of hemisphere = 2πr2+πr2=3πr22\pi r^2 + \pi r^2 = 3\pi r^2

Compound shapes

Many exam questions involve compound shapes - objects made by combining simpler 3D shapes together.

Strategy for compound shapes

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Strategy for Compound Shapes

  1. Break the shape down into recognisable parts (cones, cylinders, spheres, etc.)
  2. Calculate the surface area of each part separately
  3. Be careful about which surfaces are external (count these) and which are internal (don't count these)
  4. Add together the surface areas of all external surfaces

Practical example approach

When a hemisphere sits on top of a cylinder, the process becomes:

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Worked Example: Hemisphere on Cylinder

Step 1: Calculate the curved surface area of the hemisphere: 2πr22\pi r^2

Step 2: Calculate the curved surface area of the cylinder: 2πrh2\pi rh

Step 3: Add the area of the bottom base of the cylinder: πr2\pi r^2

Step 4: Don't count the top of the cylinder or bottom of hemisphere (they're internal surfaces)

Total surface area = 2πr2+2πrh+πr2=3πr2+2πrh2\pi r^2 + 2\pi rh + \pi r^2 = 3\pi r^2 + 2\pi rh

Exam techniques

Problem solving tips

  • Always sketch the shape if not provided
  • Label all known measurements clearly
  • Identify which formula you need before starting calculations
  • Check whether you need curved surface area or total surface area
  • Use Pythagoras when you need to find slant height

Common mistakes to avoid

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Watch Out For These Common Mistakes:

  • Confusing vertical height with slant height in cone problems
  • Forgetting to add the base area when calculating total surface area
  • Double-counting internal surfaces in compound shapes
  • Using the wrong formula for spheres vs hemispheres
  • Not using Pythagoras to find missing measurements
bookmarkSummary

Key Points to Remember:

  • Cone curved surface area = πrl\pi rl (where ll is slant height, not vertical height)
  • Cone total surface area = πr2+πrl\pi r^2 + \pi rl (base plus curved surface)
  • Sphere surface area = 4πr24\pi r^2 (four times the area of a great circle)
  • Hemisphere curved surface = 2πr22\pi r^2 (half the sphere's surface area)
  • Compound shapes require you to add surface areas of external faces only
  • Always use Pythagoras' theorem to find slant height when needed

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