You will learn
- Say what a regular polygon is.
- Use the angle sums for triangles and quadrilaterals.
- Compare regular and irregular polygons.
- Find missing angles.
Version 0.2 · expanded edition · built from cited public sources Content is based on the official frameworks and sources listed under Sources and standards. No official endorsement is claimed. Written in the style of common primary school assessments. No exam result is promised.
Student lesson
- A regular polygon has equal sides and equal angles.
- Triangle angles sum to 180°.
- Quadrilateral angles sum to 360°.
- Subtract to find a missing angle.
Words to know
All sides and angles equal.
Not regular.
Total of the interior angles.
An angle inside a polygon.
1. Discover: equal sides
A square and a rectangle both have four right angles. Which has equal sides too?
2. Understand: properties of polygons
- A regular polygon has all sides and angles equal: an equilateral triangle, a square, a regular pentagon.
- The angles of a triangle add to 180°.
- The angles of a quadrilateral add to 360°.
- Missing angle in a quadrilateral with 90°, 90°, 110°: 360 − 290 = 70°.
Idea 1: Regular
A square has equal sides and equal angles, so it is regular. A rectangle has equal angles but unequal sides, so it is irregular.
Idea 2: Triangle
60° + 60° + ? = 180°, so the missing angle is 60°.
Idea 3: Quadrilateral
100° + 80° + 90° = 270°, so the missing angle is 360° − 270° = 90°.
Why 360°? A quadrilateral splits into two triangles.
3. Worked examples
Example 1 (I show you): Regular?
Square
- Equal sides and angles.
Answer: regular
Example 2 (We do it together): Missing
60°, 60°, ?
- 180° − 120°.
Answer: 60°
Example 3 (You try): Missing
100°, 80°, 90°, ?
Show the steps and answer
- 360° − 270°.
Answer: 90°
Did you know?
Every regular polygon is equal in both sides and angles.
Why this matters
Angle sums let us find unknown angles without measuring.
4. Explore: angles
You need a protractor and paper.
- Draw a quadrilateral.
- Measure its four angles.
- Add them.
- Compare with 360°.
Think about it: what did you notice?
About 360°.
5. Practise
Say it in a sentence
Use these starters when you talk about the lesson. Say them out loud.
- The angles add to ___.
- The known angles add to ___.
- The missing angle is ___.
- It is regular because ___.
Guided practice
Question 1. Angle sum of a triangle?
180°.
Question 2. Angle sum of a quadrilateral?
360°.
Question 3. Missing angle: 90°, 90°, 110°?
70°.
Independent practice
Level 1: Start
- Is a square regular?
- Is a rectangle regular?
- Missing angle: 60°, 60°, ?
- Missing angle: 100°, 80°, 90°, ?
- Equal sides and equal angles: regular or irregular?
Level 2: Grow
- Triangle: 80°, 60°, ?
- Quadrilateral: 90°, 90°, 90°, ?
- Angle of a regular triangle
Level 3: Stretch
- Quadrilateral: 110°, 70°, 110°, ?
- Triangle with two angles of 50°: third angle
- Ben says a triangle can have angles 60°, 60° and 70°. What is wrong?
Answers are in the answer key below. Try first, then check.
6. Apply: everyday mathematics
A kite has angles 90°, 90° and 110° known. What is the fourth?
Show a sample answer
70°.
Another one. A quadrilateral garden has angles 90°, 90° and 110°. What is the fourth angle?
Show a sample answer
360° − 290° = 70°
7. Common mistakes
Fix: Regular needs equal sides too.
Fix: Triangle 180°, quadrilateral 360°.
Fix: Its sides are not all equal.
8. Lesson summary and revision card
- I can say what a regular polygon is.
- I can use angle sums.
- I can compare regular and irregular polygons.
- I can find missing angles.
9. Check and master
Answer all 14 questions. You will see feedback after each one.
Your result appears here after you answer every question.
10. Key takeaways
- Regular: equal sides and angles.
- Triangle 180°.
- Quadrilateral 360°.
11. Next lesson
Next lesson: Triangles. Keep going with the next idea in this course.
Exam practice
These questions are written in the style of common school and international primary assessments. Try them on your own, without looking back. Then check your answers in the marking guide. The marks are shown in brackets.
- Is a square regular? Explain. [2 marks]
- Is a rectangle that is not a square regular? Explain. [2 marks]
- Find the missing angle: 60°, 60°, ? [2 marks]
- Find the missing angle: 100°, 80°, 90°, ? [3 marks]
- Ali says a rhombus with four equal sides is regular. Is he right? Explain. [3 marks]
Total: 12 marks.
Marking guide and model answers
- Equal sides and angles [1]; yes [1]
- Angles equal but sides unequal [1]; no [1]
- Sum is 180° [1]; 60° [1]
- Sum is 360° [1]; 270° so far [1]; 90° [1]
- No [1]. Its angles may not all be equal [1]. It is regular only if it is a square [1].
Score guide: 11 to 12 = secure. 8 to 10 = nearly there, revise the parts you missed. Under 8 = work through the lesson again with a grown-up.
Printable worksheet
- Is a square regular? (yes/no)
- Is a non-square rectangle regular? (yes/no)
- Missing: 60°, 60°, ?
- Missing: 100°, 80°, 90°, ?
- Quadrilateral angle sum
- A regular polygon has…
- A square is…
- A rectangle (not a square) is…
- Quadrilateral angle sum:
- Missing angle: 90°, 90°, 110°:
Homework
Measure the angles of three quadrilaterals and add them.
Answer key and marking scheme
| Question | Expected response | Marks |
|---|---|---|
| Is a square regular? (yes/no) | yes | 2 |
| Is a non-square rectangle regular? (yes/no) | no | 2 |
| Missing: 60°, 60°, ? | 60° | 2 |
| Missing: 100°, 80°, 90°, ? | 90° | 2 |
| Quadrilateral angle sum | 360° | 2 |
| Total | 10 |
Practice answers: Worksheet: 1 yes; 2 no; 3 60°; 4 90°; 5 360°.
Quiz answers: 1 A, 2 B, 3 C, 4 A, 5 B, 6 A, 7 B, 8 C, 9 A, 10 B, 11 C, 12 A, Challenge 1 C, Challenge 2 C.
Extra worksheet answers (items 6 onward): 6 equal sides and equal angles; 7 regular; 8 irregular; 9 360°; 10 70°.
Independent practice answers. Level 1: 1 yes; 2 no; 3 60°; 4 90°; 5 regular. Level 2: 1 40°; 2 90°; 3 60°. Level 3: 1 70°; 2 80°; 3 They add to 190°, not 180°..
Mastery and remediation
| Objective | Bronze (Developing) | Silver (Proficient) | Gold (Advanced) |
|---|---|---|---|
| O1 Say what a regular polygon is | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O2 Use the angle sums for triangles and quadrilaterals | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O3 Compare regular and irregular polygons | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
| O4 Find missing angles | Shows this with help from an adult or a picture | Does this independently and explains how they know | Explains it clearly and uses it in a new situation |
Courseversity badge thresholds (overall quiz score): Gold = 90–100%; Silver = 80–89.9%; Bronze = 70–79.9%. A badge is never a full record of progress.
If a learner struggles
- Find the misconception: Thinking a rectangle is regular. Regular needs equal sides too.
- Explain another way: use real objects first, then pictures.
- Show a simpler example: Review angle types.
- Guided practice: work through one example together and let the learner try the next one.
- Reassess: use the quiz again or the worksheet.
- Move on when the learner shows Silver-level evidence independently.
For parents and guardians
In this lesson your child studies regular polygons and angle sums. You do not need to be good at mathematics to help.
- Spot regular shapes at home.
- Ask: are all sides and angles equal?
- Cut a quadrilateral into two triangles.
- Ask “how do you know?” after each answer, and listen.
- Praise effort, not only right answers. Mistakes are how we learn.
- Ten to fifteen minutes a day is better than one long session.
Teacher package
1–5. Objectives, prerequisites, materials, preparation, duration
Lesson objectives. Learners will: say what a regular polygon is; use the angle sums for triangles and quadrilaterals; compare regular and irregular polygons; find missing angles.
Prerequisite knowledge. Lesson 1.
Materials. Counters (stones, seeds or bottle tops), paper, pencils, board or chalkboard. Low-resource options: use stones, seeds, sticks, bottle tops or drawings in the sand or on the board instead of printed materials.
Preparation. Gather the counters and draw the example pictures on the board before the lesson.
Suggested duration. 45–60 minutes. It can be split into two shorter sessions.
6–9. Teaching procedure, explanation notes, guided and independent activities
Introduction (5 min). Start with the Discover situation and let learners suggest ideas before you teach.
Explanation (10 min). Teach the Understand section with real objects first, then pictures, then symbols and words.
Guided activity (10 min). You need a protractor and paper. Learners follow the Explore steps in pairs.
Independent activity (10 min). Learners complete the independent practice and the worksheet.
Assessment and reflection (5 min). Ask each learner to say one thing they learned and how they know it.
Key questions. Are sides and angles equal? What do the angles add to?
Teacher explanation notes. Use the concrete, pictorial, abstract order: real objects first, then pictures, then symbols and words. Return to real objects whenever a learner needs support.
10–12. Differentiation, formative assessment, common misconceptions
Support. Use real objects and work with a partner; say each step aloud together. Extension. Ask learners to make up their own example and explain it to a partner.
Formative assessment. Listen to learner explanations, ask “how do you know?”, and check the independent work.
Misconceptions. Thinking a rectangle is regular. Regular needs equal sides too.
13–15. Assessment questions, answer keys, marking scheme
Use the quiz and worksheet on this page. The 12-mark exam practice scheme, the 10-mark scheme and all answers are in the answer sections above. Re-check every answer and the marking scheme before publication.
16–19. Remediation, enrichment, homework, teacher reflection
Remediation. Review angle types. Enrichment. Invite learners to find or create three more examples at home or in the classroom and explain them. Homework. See above.
Teacher reflection. After teaching, record: which concepts learners understood; which misconceptions occurred; which activities worked; which learners need more practice; and what to improve next time.
Sources and standards
This lesson is based on the sources below, and no official endorsement is claimed. Reference starting points named in the Courseversity standard: Common Core State Standards for Mathematics, Cambridge Primary and the IB Primary Years Programme. Where a standard code is named, it points to the official framework.
Official standards for this topic area. Common Core State Standards for Mathematics (official site): Grade 5 » Geometry. These pages list the standards that cover this topic area; read them alongside the lesson.
Research used to shape this lesson. Common Core 5.G.B.3 and 4.MD.C.7; Cambridge Primary Mathematics Stage 5 and 6 (polygon properties). These sources informed the lesson.