National 5MathsSQAApplications of MathematicsRevision NotesGeometric SkillsPerimeter & AreaPerimeter & Circumference

Perimeter & Circumference (SQA National 5 Applications of Mathematics): Revision Note

Exam code: X844 75

Dan Finlay

Written by: Dan Finlay

Reviewed by: Roger B

Updated on

Perimeter

What is perimeter?

  • Perimeter is the total distance around the outside of a 2D shape

  • Perimeter is a length in one dimension

    • Units of measure include mm, cm, m, etc.

How do I find the perimeter of a 2D shape?

  • Add together the lengths of all the sides of the shape

  • There might be some sides which have the same lengths

    • These are marked with dashes to indicate which sides are equal

How do I find the perimeter of a composite shape?

  • Composite shapes are made up of two or more 2D shapes

    • Composite shapes can usually be split into rectangles, triangles and parts of circles

  • Compare vertical and horizontal lengths separately

    • You might need to add lengths to find missing longer lengths

    • You might need to subtract lengths to find missing shorter lengths

Illustration of an L-shaped area with dimensions: 10m and 15m on top, 16m and 11m on the bottom, dividing it into rectangles with calculations shown.
Example of finding missing lengths

Worked Example

Find the perimeter of the logo shown below.

Geometric shape with measurements: 18 cm length, 6 cm angled section, 2 cm width, 15 cm base, and 4 cm height on the right side.

Answer:

The shape can be split up into a triangle and two rectangles

  • The dashes on the triangle mean that the lengths are equal

  • The missing length is 6 cm

The horizontal lengths of the two rectangles add up to the length of the longest horizontal line

  • Subtract the horizontal lengths to find the missing horizontal length

18 minus 15 equals 3 space cm

Geometric shape with a rectangle measuring 18 cm by 6 cm, a triangle on the left, and a right-angle extension on the right, with various dimensions marked.

Add the lengths of all the sides to find the total perimeter

6 + 6 + 18 + 2 + 3 + 4 + 15 = 54

54 cm

Circumference of a circle

What is the circumference of a circle?

  • The circumference of a circle is its perimeter

How do I find the circumference of a circle?

  • You are given the formula C equals pi d

  • Identify the diameter

    • This is double the length of the radius

  • Multiply the diameter by π

Diagram of a circle with a dashed line showing the diameter. Formula for circumference, C = πd, is displayed below the circle.
Circumference of a circle

How do I find the perimeter of composite shapes involving parts of circles?

  • Identify the diameter of the full circle

    • You might have to double the radius if there is a quarter circle

  • Calculate the circumference of the full circle

  • Divide the circumference by:

    • 2 if it is a semicircle

    • 4 if it is a quarter circle

  • Remember to add the lengths of any straight sides when calculating the perimeter

Diagram showing half and quarter circles. Left: semicircle with diameter, curved length πd/2. Right: quadrant with radius, curved length πd/4.
Parts of circles

Examiner Tips and Tricks

You can put parts of circles together to make your calculation easier. For example, if the shape contains two equal semicircles, then you can just find the circumference of the full circle. This gives the same answer as halving it (to find the length of the curved side of one semicircle) and then doubling it.

Worked Example

Jim has a garden in the shape of a square and two quarter circles as shown below.

Dome-shaped semicircle with two vertical dashed lines inside, base measuring 4.2 metres, indicating dimensions or architectural feature.

Jim wants to put a fence all around the outside of the garden. Calculate the length of fence required. Round your answer to the nearest metre.

Answer:

The radius of the quarter circle is equal to the length of a side of the square

The length along the bottom is made up of three equal lengths:

  • the radius of the first quarter circle

  • the length of a side of the square

  • the radius of the second quarter circle

Divide the length by three to find the length of a side of the square and the radius

4.2 space straight m divided by 3 equals 1.4 space straight m

Find the length of the curved sides

  • Find the diameter of the full circle

1.4 space straight m cross times 2 equals 2.8 space straight m

  • Find the circumference of the full circle

pi cross times 2.8 equals 8.796...

  • Divide by two as the two quarter circles make a semicircle

8.796... divided by 2 equals 4.398...

Add together:

  • the total curved length

  • the long length of the bottom

  • the straight length of the square at the top

4.398... plus 4.2 plus 1.4 equals 9.998...

Round to the nearest metre

10 m

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    Author
    Reviewer
    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

    Roger B

    Reviewer: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.