Angles in a polygon

Split a polygon into triangles from one corner. A shape with n sides makes n − 2 triangles, so its angles add to (n − 2) × 180°.

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.

In the last lesson you found that the angles of any triangle add to $180°$ . A polygon is any closed shape with straight sides: a quadrilateral (4 sides), pentagon (5), hexagon (6), octagon (8) and so on.

In this lesson you use the triangle rule to find the total of a polygon's angles, without measuring them.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

Four corners

Think about a square or a rectangle first. Does a lopsided quadrilateral have the same total?

What do the four angles of a quadrilateral add to?

540°

180°

, like a triangle

360°

It depends on the shape

TryTry it on the board and compare the result with your prediction.

Split a quadrilateral

The board measures each angle and adds them up. Split into triangles draws lines from corner $A$ to the corners that are not next to it. Drag the corners and watch both totals.

Try it yourselfYour edits are saved in this browser.

Press **Split into triangles**, then drag the corners into a new quadrilateral. Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

A triangle for each side, less two

Here is the method for a pentagon.

Worked exampleBefore each step, say what you think comes next.

$\text{A pentagon has 5 corners}$ Choose one corner, $A$ .
$\text{Lines from A to C and to D}$ Join $A$ to every corner that is not next to it.
$3 \text{ triangles}$ The two lines split the pentagon into 3 triangles.
$\text{Their angles fill the pentagon's corners}$ Every triangle's corners are at the pentagon's corners, and together they fill them exactly.
$3 \times 180° = 540°$ So the pentagon's angles add to 540°.
$(n - 2) \times 180°$ A shape with $n$ sides splits into $n - 2$ triangles this way.

Read Ben's working one line at a time. Which line counts angles that are not the pentagon's?

Try it yourselfYour edits are saved in this browser.

Split shapes on the board, or in your head, and fill in the table.

sides	3	4	5	6	8
triangles from one corner	1
angle sum (°)	180

Ben finds the angle sum of a pentagon. Click the line where his reasoning goes wrong.

What do the angles of a hexagon (6 sides) add to?

ModifyChange the problem on purpose and see what happens.

One change at a time

Each task changes one thing. First: the same pentagon in a new shape.

Need a hint?

Move any corner. Does the number of triangles change?

Need a hint?

Drag one corner towards the middle of the shape, past the line joining its neighbours.

Another hint

Look at the angle at that corner: it is more than 180°.

Need a hint?

Use More sides. How many triangles does each new side add?

Try it yourselfYour edits are saved in this browser.

Drag the pentagon into a lopsided shape (keep every corner pointing out). Split it: what is the total?

Use the board. This ticks when you get there.

Edge case: drag one corner in so it points into the shape. What is the total now?

Use the board. This ticks when you get there.

Make a polygon whose angles add to $1080°$ .

Use the board. This ticks when you get there.

Without the board: what do the angles of a 12-sided polygon add to?

PractiseA few on your own, to check you have it.

The rule

What do the angles of a 10-sided polygon add to?

A regular hexagon has 6 equal angles. How big is each one?

A polygon's angles add to $900°$ . How many sides does it have?

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

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

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 $1000°$ ?

Try it yourselfYour edits are saved in this browser.

Need a hint?

How many lots of 180° make 1260°?

Another hint

That is the number of triangles. A shape has 2 more sides than triangles.

Make a polygon whose angles add to $1260°$ .

Use the board. This ticks when you get there.

For teachers

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). About 45 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

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.

Before class

Do the Run board and the Make task yourself first. Students need the last lesson: a triangle's angles add to 180°.

The method is to draw lines from one corner only (the board's Split into triangles). Students who split from a point in the middle get n triangles and must take off the full turn of 360° round that point: that is Ben's mistake, planted after the method.

The board only splits convex shapes (every corner points out). A concave shape (a corner that points in, with a reflex angle) is the edge case in Modify: the angle sum is still (n − 2) × 180°.

Running the lesson

Predict: take a show of hands. Students who chose 180° are treating every shape like a triangle; '4 × 90°' students may pick 360° for the wrong reason. Ask: does a lopsided quadrilateral still add to 360°?
Run: ask students to drag the corners into a very lopsided quadrilateral and read both totals: the measured one and 2 × 180°.
Investigate: model the worked example live. Draw a pentagon, choose A, draw the two lines at writing pace, shade the 3 triangles, then write 3 × 180° = 540°. Reveal the prepared lines afterwards to compare. Then fill the table together and ask: what is the pattern?
Modify: before the 1080° task ask students to predict the number of sides, then use More sides to check.
Practise: the 'backwards' question plants 5 (the number of triangles). Ask: how many sides make 5 triangles?
Make: collect answers and reasons. Then the stretch: 1000 is not a multiple of 180.

When students get stuck

A student counts n triangles (Ben's split from the middle). Ask: Which angles in your triangles are not corners of the polygon? Fix: Point at the angles round the middle point: a full turn, 360°. Take it off, or split from a corner instead.
A student draws lines from A to the corners next to it. Ask: Do those lines make a new triangle? Fix: Those lines are sides. Join A only to the corners that are not next to it.
A student thinks each angle of any pentagon is 108°. Ask: Is the pentagon on the board regular? What happens to one angle when you drag a corner? Fix: 108° is each angle of a regular pentagon only: 540° shared equally. The total is 540° for every pentagon.

Stretch and support

Stretch: Why can no polygon have angles adding to 1000°? Each interior angle of a regular polygon with 20 sides? (162°.)

Support: Use the choose board (More sides) to count triangles for each shape. A printed table with the first two columns filled helps students see the pattern.

Where it leads

Next in class: finding unknown angles in triangles and polygons, giving the angle rule as the reason (node 3).

Where it leads: exterior angles (they add to 360° for any convex polygon), which regular polygons tessellate, and the turtle: a closed shape turns 360° in total, which the coding pathway uses for its polygons.

What each step is for

four corners: Committing to a total for a quadrilateral before splitting it makes the board a test of their idea: 4 corners do not mean 4 × something, or 180° again.
split a quadrilateral: Splitting a quadrilateral from one corner shows two triangles whose angles exactly fill the four corners, so the total is 2 × 180° whatever the shape.
a triangle for each side, less two: Lines from one corner split a shape with n sides into n − 2 triangles, and those triangles' angles exactly fill the polygon's corners, so the sum is (n − 2) × 180°. Ben's split adds the angles around his middle point too.
one change at a time: One change at a time: a lopsided pentagon (same total), a corner that points in (still 360°), more sides on the board, then the rule without the board.
the rule: The rule forwards, for an equal-angled shape, and backwards from a total, mixed; the review keeps the triangle rule fresh.
from a total to a shape: Building a shape from a given total reverses the lesson: from the angle sum back to the number of triangles and sides.

Now you can

Split a polygon into triangles from one corner and find its angle sum.
Explain why a shape with n sides has an angle sum of (n − 2) × 180°.
Find the number of sides from an angle sum.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson6. Unknown angles in shapesFind unknown angles in triangles and polygons in steps, with a reason for each: the angle sum of the shape, angles on a straight line, equal angles in an isosceles triangle. An exterior angle of a triangle equals the sum of the two interior opposite angles.

Source: content/maths/angles/polygon-angles/lesson.md