Worked examples
The question
Question: find the sum of the interior angles of a pentagon.
1 Cut into triangles
A pentagon has 5 sides and 5 corners.
A polygon is a flat shape with straight sides.
Interior angles are the angles inside, at the corners.
Pick one corner.
Draw lines from it to the other corners.
They cut the pentagon into triangles.
Count them: 1, 2, 3.
Each triangle sits on one side of the pentagon, away from our corner.
The two sides that touch our corner have no triangle of their own.
So there are 2 fewer triangles than sides: 5 − 2 = 3.
triangles = 5 − 2 = 3
2 Add the triangles
The angles in each triangle add up to 180°.
Together, the triangles make all the angles of the pentagon.
3 triangles, 180° each: 3 × 180.
sum = 3 × 180
Careful: we count the triangles, 3, not the sides, 5.
not 5 × 180!
3 times 180 is 540.
So the angles add up to 540°.
sum = 540°
3 Check
Let's check: add the 3 triangles one by one.
180 + 180 + 180 = 540
We get 540° again.
So the sum is right!
Let's look back.
First: cut into triangles.
Then: add the triangles.
Last: check.
▶ Watch it solved
The question
Question: find one interior angle of a regular pentagon.
1 Cut into triangles
A pentagon has 5 sides and 5 corners.
A polygon is a flat shape with straight sides.
Interior angles are the angles inside, at the corners.
Regular means all the sides are equal, and all the angles are equal.
Pick one corner.
Draw lines from it to the other corners.
They cut the pentagon into triangles.
Count them: 1, 2, 3.
Each triangle sits on one side of the pentagon, away from our corner.
The two sides that touch our corner have no triangle of their own.
So there are 2 fewer triangles than sides: 5 − 2 = 3.
triangles = 5 − 2 = 3
2 Add the triangles
The angles in each triangle add up to 180°.
Together, the triangles make all the angles of the pentagon.
3 triangles, 180° each: 3 times 180 is 540.
sum = 3 × 180 = 540
3 Share it out
All 5 angles are the same.
So we share 540° equally: 540/5.
one angle = 5405
540 divided by 5 is 108.
So each angle is 108°.
one angle = 108°
4 Check
Let's check: 5 angles of 108° together.
5 × 108 = 540
We get 540°, the angle sum.
So 108° is right!
Let's look back.
First: cut into triangles.
Then: add the triangles.
Then: share it out.
Last: check.
▶ Watch it solved
The question
Question: find one interior angle of a regular heptagon.
1 Cut into triangles
A heptagon has 7 sides and 7 corners.
A polygon is a flat shape with straight sides.
Interior angles are the angles inside, at the corners.
Regular means all the sides are equal, and all the angles are equal.
Pick one corner.
Draw lines from it to the other corners.
They cut the heptagon into triangles.
Count them: 1, 2, 3, 4, 5.
Each triangle sits on one side of the heptagon, away from our corner.
The two sides that touch our corner have no triangle of their own.
So there are 2 fewer triangles than sides: 7 − 2 = 5.
triangles = 7 − 2 = 5
2 Add the triangles
The angles in each triangle add up to 180°.
Together, the triangles make all the angles of the heptagon.
5 triangles, 180° each: 5 times 180 is 900.
sum = 5 × 180 = 900
3 Share it out
All 7 angles are the same.
So we share 900° equally: 900/7.
one angle = 9007
900 divided by 7 does not come out even: 128.57 and more digits.
We round it to 128.6.
900 ÷ 7 = 128.57…
So each angle is about 128.6°.
one angle ≈ 128.6°
4 Check
Let's check: 7 angles of 128.6° together.
7 × 128.6 = 900.2
We get about 900°, the angle sum.
It is a little off because we rounded.
So about 128.6° is right!
Let's look back.
First: cut into triangles.
Then: add the triangles.
Then: share it out.
Last: check.
Answer one angle ≈ 128.6°
▶ Watch it solved
