Angles on Polygons | Maths GCSE
Вставка
- Опубліковано 10 вер 2024
- This lesson explains how to solve maths GCSE problems involving angles on polygons.
Here are the key points to remember:
- The external angles of polygons always add to give 360 in total.
- You can find an internal angle's size by subtracting the external angle from 180.
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Sum of angles of a polygon is (n-2) * 180. For a regular polygon, the sides and interior angles have to be equal, so you can compute x as (5-3)*180/5 = 108.
Or just 540 degr. / 5 = 108 degr.
The pentagon can be divided into 3 adjacent triangles, which each contributes 180 degr. - which adds up to 540 degr.,
Well done. The simplest math to do …..thxxx …..
Or if all the sides are equal then use (n-2)×180 and divide it by the number of sides
Except that doesn’t explain why it works. The best way to learn is to understand rather than remember.
@@ChrisLee-yr7tz well (n-2)x180 is easier to remember and explain because (n-2) is the number of triangles the polygon can be split into and each angle sum of a triangle is 180 so it’s much easier to calculate and understand
@Cookie Derp That's fine. It wasn't my point.
My point was that when you're teaching maths it's important to understand why you're doing something and why it works.
Just saying remember this formula and use it is very bad. It's pointless.
@@ChrisLee-yr7tz I just explained how it worked (n-2) is the number of triangles that the polygon can be dissected into and the sum of all angles of a triangle is 180 degrees
@@cookiederp3573 I know why it works and I know you (partly) explained why it works.
I’ll say it again: Several people made the point of ‘just use (n-2)*180’. My point was that it’s not useful in general just saying ‘use this formula’ which is what the OP did. The fact you went on to explain re the triangles wasn’t relevant to my point. My point is a general point re understanding derivations. It’d be like just telling someone that when you differentiate x^2 you just use 2x. It’s pointless.
Even your point isn’t a general solution or an explanation. Why do the angles of a triangle add up to 180? All you’ve done is reduce a polygon down to a series of triangles and just assumed that a triangle adds up to 180.
Prove they do!
The real explanation is to walk around a polygon and count the degrees you turn at each corner. To walk around the shape you have to make one complete turn, ie 360 degrees.
So each exterior angle is 360/n in a regular polygon.
The internal angles are therefore 180 - 360/n
This can be rearranged to 180(n-2) / n.
interior angles of a polygon = 2n - 4 right angles. So 10 - 4 = 6 right angles. Therefore 540 degrees / 5 = 108 degrees in each corner of the regular pentagon.
Measure of each angle in a regular polygon =180-360/n degrees
Derivation:-
Measure of sum of angles of regular polygon=(n-2)180,where n is number of sides of regular polygon
=180n-360 degree
Measure of each angle
=(180n-360)/n
=180-360/n degree
Since here it's a regular pentagon , which has 5 sides,
Measure of each degree
=180-360/5
=180-72
=108°
This derivation can be used to prove that the number of sides of regular polygon is directly proportional to measure of each of its angles
As 180-360/n increases as n increases as the value of 360/n decreases for successive values of n where n≥1
That's not a derivation
It would have been nice to demonstrate how we know the external angles add to 360 degrees, instead of skimming over that introductory fact. Maybe it’s very obvious but I would have enjoyed having the reason pointed out. Other than that, good show.
A useful way I found with explaining that to my kids is to imagine walking around the shape..(actually we did it for real on the floor). It becomes obvious that your body does one full turn to get back to the starting position.
Draw a line between two angles. One part will be triangle and one will be quadrangle. Total angles of a triangle is 180 and total angles of a quadrangle is 360 so Total angles of pentagon is 540, divided by 5 = 108
EXCELLENT! So easy and so clever. Best thing here. Should get multi-K likes, not just my first “1”.
Thanks
I learned from my school
Triangle
Quadrilateral
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Hendegon
Dodecagons
Nice tip!quick and easy
tnx
180•3/5= 36•3= 108
😂
Sweet
Regular shapes have like properties?
Regular pentagon= 108°
Regular hexagon= 120°
Regular octagon= 135°
I just learnt it like that at junior high school or something
How to find a problem to solve when you don't have any
It’s so sad the number of comments just saying use (n-2)x180 and divide by the number of sides.
The whole point of the video is to understand the geometry of how to calculate the interior angle.
You all also seem to have missed the fact that how he calculated it is actually the derivation of your method in the first place! It’s the same formula ffs!!!
Your answers:
(n-2)x 180 / n
Video method:
180 - (360 / n)
=180n/n - 360/n
=(180n - 360 )/n
=180(n-2)/n
=(n-2)x 180 / n
Learn to understand maths and you don’t ever need to remember anything..
Btw - I also found that when I taught my kids this method it was useful to tell them to imagine walking around the shape, keeping track of their direction in their head. That makes it obvious to a young child why the external angles add up to 360 degrees when they realise they do a single turn once they end up back where they started.
Sir, I need to find out area of a quadrilateral whose all 4 sides are given but none of the angles is known. Can u help please?
If yes and you plan to respond to this query through a video then pls do reply to this comment because that'll notify me to check your video.
Working on a video for you!
If you want to send me a link to the exact problem you need help with feel free to reply!
@@revisionboost I don't know how to create a link. The question is in my head. Like whenever for a quadrilateral (lets say a real estate plot) we know all the 4 sides dimensions but dont know the exact angles, then how can we rescale it for working because if angles are known, i can replicate the drawing on a bigger scale
Consider a plot in a residential society whose all 4 sides are known and thats it we know, angles are not confirm because they are not 90 degree
Can I send u a picture if you have an email address etc?
Don’t all pentagon angles equal 108 degrees.
A general polygon with N corners has an internal angle sum of (N-2) × 180°.
So if all angles are the same you get:
(N-2) × 180° ÷ N =
N × 180° ÷ N - 2 × 180° ÷ N =
180° - 360° ÷ N
How to prove the general formula I started with?
Quite easy by induction:
The internal angle sum of a triangle is 180° and
every polygon with N corners is also a polygon with N+1 corners where the extral corner has an internal angle of 180°.
yeah, and they do in this video too. why did you ask
Not all pentagons. But _regular_ pentagons do; it's a basic, but important distinction.
No
@@mittelwelle_531_khz You prove (n-2)x180 / n by just rearranging the formula of what he did in the video!
Now why do external angles add up to 360 degrees?
(5-2)*180/5 is x
3*180/5= 108
Si tuviesemos un punto en el centro del pentágono regular al unirlo a los vértices tendríamos cinco triangulos idénticos 360÷5= 72°
La suma de los 3 ángulos de un triángulo es 180° y por ser todos los triángulos iguales la suma de uno de los ángulos con su inmediato del otro triángulo será 180° - 72° = 108°
Or (n_2)×180 ÷5 =108
i just followed the formula: 180(n - 2)
then divided it by 5
60
"External angles always add up to 360 degrees."
Giving proof of that is much more interesting than your calculation for a pentagon.
This only works for regular polygons unfortunately there are lots of problems out there that deal with irregular polygons.
X=180-(360/5)=180-72=108
You neglected to state what a regular pentagon is or what the properties are.
540/5=108
X=3×180/5=3×36=108
The way you get the answer is not clear 😢
Pentigun
It’s not x it’s alpha ❤
x
I understood everything until he started talking
Geometry Geometry 😓😓
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