Circle and Cylinder Calculations
This topic covers calculating the dimensions of 2D and 3D shapes. It's a frequent and important ESAT topic that tests your ability to apply geometric formulae accurately to both simple and complex 'composite' shapes without a calculator.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- You must memorise the formulae for the area and circumference of a circle, and the volume of a cylinder. Formulae for spheres, cones, and pyramids will be provided in the exam.
- Composite shapes are made by adding or subtracting simpler shapes. Their properties are found by combining the corresponding formulae.
- The perimeter of a composite 2D shape is the length of its external boundary only. Do not include lines where the simple shapes are joined.
- The surface area of a composite 3D solid excludes the surfaces where the simple solids are joined together.
- Be ready to use Pythagoras' theorem to find missing lengths, such as the slant height of a cone, before applying a surface area or volume formula.a2 + b2 = c2
Formulae
Circumference = 2 × pi × r To find the length of the boundary of a full circle. For a semi-circle or other arc, use the appropriate fraction of this.
Area = pi × r2 To find the space enclosed by a full circle. Remember to use the radius, not the diameter.
Volume of Cylinder = pi × r2 × h To find the space inside a right circular cylinder, where 'h' is the perpendicular height.
Definitions
- Composite Shape
- A 2D or 3D figure formed by combining two or more basic geometric shapes.
- Perimeter
- The total length of the continuous line forming the outer boundary of a closed 2D shape.
- Surface Area
- The total area of all the exposed faces and curved surfaces of a 3D object.
Worked example
A running track is formed by a rectangle of length 100 m and width 50 m, with a semicircle attached to each of the 50 m ends. What is the total perimeter of the track in metres?
- 1
Identify the components of the perimeter.
The perimeter consists of the two straight sections and the two curved sections.
- 2
The two straight sections correspond to the 100 m sides of the rectangle.
Their total length is 2 × 100 = 200 m.
- 3
The two semicircles at the ends together form one full circle.
The diameter of this circle is the width of the rectangle, which is 50 m.
- 4
Calculate the circumference of this full circle.
First, find the radius:
r = diameter / 2 = 50 / 2 = 25 m - 5
The circumference is C = 2 × pi × r = 2 × pi × 25 = 50 × pi metres.
- 6
Add the lengths of the straight and curved parts to find the total perimeter:
200 m + 50 × pi m.
Answer: 200 + 50 × pi
Common mistakes
- ×Confusing radius and diameter. Questions often give the diameter to trap you. Always halve it to get the radius 'r' before using formulae like `pi × r2`.
- ×Forgetting fractional multipliers. Remember to use 1/2 for semicircles and hemispheres, and watch for the 1/3 factor in the given cone and pyramid volume formulae.
- ×Incorrectly calculating composite perimeters or surface areas. Only include the outer boundary for perimeter and exposed surfaces for surface area. The joined edges/faces are not included.
- ×Arithmetic errors with pi. Keep pi as a symbol throughout your working and only combine the numerical coefficients. Avoid trying to approximate its value.
No-calculator tips
- ✓Always leave pi as a symbol (π) in your working. Simplify all other numbers first. This avoids messy arithmetic and makes cancellation easier.
- ✓When dealing with ratios of areas or volumes, use scaling factors. If a length is scaled by a factor 'k', its area scales by k2 and its volume scales by k3. For example, doubling the radius of a sphere increases its volume by 23 = 8 times.
- ✓Factor out common terms before calculating. For a composite volume like a cylinder with a cone on top, you can factor out `pi × r2` to simplify the addition.