Figurate Numbers
Square, pentagonal and other numbers arranged as shapes.
Key idea
Square, pentagonal and other numbers arranged as shapes.
In depth
Figurate numbers count dots arranged in regular shapes: triangular, square (n²), pentagonal, and so on. They link counting to geometry and show how sequences relate: consecutive triangulars sum to squares.
Worked example
Square numbers 1, 4, 9, 16. Triangular: 1, 3, 6, 10. T(3) + T(4) = 6 + 10 = 16 = 4².
Real-world uses
- Recognising patterns in tiling and arrangement puzzles
- Introducing sequences and formulas in class
Common mistake
Mixing up which shape a formula belongs to.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Figurate Numbers?
Square, pentagonal and other numbers arranged as shapes. It is a topic at the advanced level in Arithmetic & Number Sense.
Where is Figurate Numbers used in real life?
Real-world examples: Recognising patterns in tiling and arrangement puzzles; Introducing sequences and formulas in class.
What should I learn before Figurate Numbers?
Make sure you are comfortable with Triangular Numbers, the previous topic in Arithmetic & Number Sense.
What is a common mistake with Figurate Numbers?
Mixing up which shape a formula belongs to.
How can I practise Figurate Numbers?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Figurate Numbers?
Move on to Arithmetic Sequences (Intro) next.
Related advanced topics in Arithmetic & Number Sense
- Repeating Decimals – Decimals with a repeating block are rational and convertible to fractions.
- Compound Interest – Interest earned on principal and past interest.
- Inverse Proportion – When one quantity doubles, the other halves.
- Rational Numbers – Numbers expressible as p/q with integers p, q, q≠0.
- Irrational Numbers – Real numbers that cannot be written as a fraction, such as √2 and π.
- Real Numbers – All rational and irrational numbers; the continuous number line.