--- title: "Angles in a polygon" subtitle: "Angles and shapes · Angle sums · Lesson 5" --- # Angles in a polygon **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Stop signs are octagons, honeycomb cells are hexagons, a soccer ball is pentagons and hexagons. The triangle rule tells you the angles of all of them. :::teacher-note **In one sentence:** Students split polygons into triangles from one corner, find that a shape with n sides makes n − 2 triangles, and use (n − 2) × 180° for the angle sum (AC9M7M05, node 2). **Students should be able to say:** I can split a polygon into triangles from one corner and find its angle sum. I can explain why a shape with n sides has an angle sum of (n − 2) × 180°. I can find the number of sides from an angle sum. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict Think about a square or a rectangle first. Does a lopsided quadrilateral have the same total? :::keep-together **What do the four angles of a quadrilateral add to?** ☐ $540°$ ☐ $180°$, like a triangle ☐ $360°$ ☐ It depends on the shape My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 360°. Any quadrilateral splits into 2 triangles from one corner, and 2 × 180° = 360°. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## A triangle for each side, less two :::keep-together ### Splitting from one corner Here is the method for a pentagon. 1. $\text{A pentagon has 5 corners}$ — Choose one corner, $A$. 2. $\text{Lines from A to C and to D}$ — Join $A$ to every corner that is not next to it. 3. $3 \text{ triangles}$ — The two lines split the pentagon into 3 triangles. 4. $\text{Their angles fill the pentagon's corners}$ — Every triangle's corners are at the pentagon's corners, and together they fill them exactly. 5. $3 \times 180° = 540°$ — So the pentagon's angles add to 540°. 6. $(n - 2) \times 180°$ — A shape with $n$ sides splits into $n - 2$ triangles this way. **Split shapes on the board, or in your head, and fill in the table.** {{rows 1}} | sides | 3 | 4 | 5 | 6 | 8 | |---|---|---|---|---|---| | triangles from one corner | 1 | | | | | | angle sum (°) | 180 | | | | | ::: :::teacher-note **Answer:** 2, 3, 4, 6, 360, 540, 720, 1080 Each shape makes 2 fewer triangles than it has sides, and each triangle adds 180°. ::: :::keep-together ### Find the mistake **Ben finds the angle sum of a pentagon. Click the line where his reasoning goes wrong.** ``` 1 Ben wants the angle sum of a pentagon. 2 He puts a point in the middle and draws a line to each corner. 3 That makes 5 triangles. 4 5 × 180° = 900°, so the pentagon's angles add to 900°. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 4 Line 4: his 5 triangles also have angles around the middle point, a full turn of 360°, which are not angles of the pentagon. 900° − 360° = 540°. ::: :::keep-together ### A hexagon **What do the angles of a hexagon (6 sides) add to?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 720 6 − 2 = 4 triangles, and 4 × 180° = 720°. ::: :::typical ## One change at a time **A lopsided pentagon.** Each task changes one thing. First: the same pentagon in a new shape. **A corner that points in.** **More sides.** **Twelve sides.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## The rule :::keep-together ### Ten sides **What do the angles of a 10-sided polygon add to?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 1440 (10 − 2) × 180° = 1440°. ::: :::keep-together ### Equal angles **A regular hexagon has 6 equal angles. How big is each one?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 120 The angles add to 720°, so each is 720° ÷ 6 = 120°. ::: :::keep-together ### Backwards **A polygon's angles add to $900°$. How many sides does it have?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7 900° ÷ 180° = 5 triangles, and a shape with 7 sides makes 7 − 2 = 5 triangles. ::: :::keep-together ### Review **Review: a triangle has angles $35°$ and $75°$, and a third angle. Do the three add to more than, less than, or exactly $180°$?** ☐ Exactly $180°$ ☐ Less than $180°$: $35 + 75 = 110$ ☐ It depends on the third angle ::: :::teacher-note **Answer:** Exactly $180°$ Every triangle's angles add to 180°, so the third angle is 180° − 110° = 70°. ::: ## Make: from a total to a shape Work it out first, then use **More sides** to check. Stretch: why can no polygon have angles that add to $1000°$? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Find the angle sum of a heptagon (7 sides), and explain your method.** {{lines-3}} ::: :::teacher-note **Answer:** 900°. From one corner it splits into 7 − 2 = 5 triangles, and 5 × 180° = 900°. :::