LED strip project questions

I'm making a present for someone that is basically a frame outlined in LEDs and then their name is written in LED strips in the middle. The Arduino is to control a constant color change in the LEDs. The problems(at least the ones I'm asking about here) are that the LED strip I purchased takes 1.5amps at 24v, which I'm pretty sure will fry my Arduino. And that it can only be cut every 15 LEDs which means the shortest strip I could get is 25cm which is too long. (this is the product info on Amazon for reference if that helps:
https://www.amazon.com/gp/product/B075FGJXQ7/ref=oh_aui_detailpage_o01_s00?ie=UTF8&psc=1 )

I am wondering if there is any way to power the Arduino and the lights from the same power source. Is there some kind of circuit I could build or a component I could buy that would let the Arduino run without dying and still control the LEDs? And is there any way I could cut smaller strips than every 15 LEDs that would still be able to work? I'm not too technologically knowledgeable but I would at least consider myself proficient. I definitely won't know all of the terminologies so an in-depth explanation or solution would be greatly appreciated.

WS2812 LED strips would be much easier to use. Can be cut to any number of LEDs, work on 5V and take just 1 output pin.

Aode_Ward:
it can only be cut every 15 LEDs which means the shortest strip I could get is 25cm which is too long.

Sounds like you bought the wrong product for your project. You might want to use a different one that allows you to size the strips to the correct length.

Aode_Ward:
Is there some kind of circuit I could build or a component I could buy that would let the Arduino run without dying and still control the LEDs?

The 24 V is too high for the Arduino's voltage regulator. You can get a step down module that will reduce the 24 V to a voltage your Arduino can use. These modules are widely available and quite cheap. Something like this:
https://www.amazon.com/dp/B07CYNHC46
There is a potentiometer you use to set the output voltage. You will need a voltmeter to check what you have it set to. So you would use a 24 V power supply connected directly to the LED strip, then the step down module connected between the 24 V power supply and the Arduino board.

Aode_Ward:
is there any way I could cut smaller strips than every 15 LEDs that would still be able to work?

I'm sure there is, but I couldn't say what needs to be done without looking at how the strip is wired. You can buy strips that are made to be cut in smaller increments. Buying a more suitable strip would also allow you to get one with a lower working voltage, which would permit you to power the Arduino directly from the power supply (via the voltage regulator on the Arduino board if over the working voltage of the Arduino), with no need for the step down.

With a 24V strip there will be a lot of LEDs in series for each section, making the sections between the
cut-points much longer - simple fact of life. The plus side is you get longer strip runs between power
feed points.

HI,Aode_Ward, I think you can use the IC ,such as th LM7805, to change the 24V to 5V. The 5V power supply can be used for Arduino. Then use the Arduino to control the triodes(for example, 8550 ) and the triodes drive the LED srtip. The circuit is very simlpe.

I did some tests on some lighting products, such as led backlight panel, led panel light and so on. This method always works.

That strip alternates red, green and blue. I did wonder how you could drive 15 LEDs from only 24V, but it is that you have 5 LEDs of each colour in seriese.
I think the illustration of each light being a different colour is designed to trick you into thinking that these are addressable strips, which they are not.

You could cut the strips into smaller sections but then you would have to wire in another resistor and use a different voltage to drive them. You could cut them into strips of three but always one led would be red then next green and the final one blue which is not what you want.

The Arduino is to control a constant color change in the LEDs

The LEDs are too far apart for any change in colour due to light mixing, all you could do is to dim the brightness of each colour LEDindividually.

blueskysea:
HI,Aode_Ward, I think you can use the IC ,such as th LM7805

A linear voltage regulator like LM7805 is going to get pretty hot with a 24 V supply. That's why I recommended the use of an efficient switching module.

So as it has been pointed out that the LEDs I have are probably not the best suited for this project and I was helpfully directed to these LEDs. which of the variants would be recommended for lettering?

I was going to refer you to where i did my last purchase, of side-emitting LED's 5v - 60 LED's & chips p/m but that site is gone.. anyway side emitting are the best for flowing curved letters, but quite expensive at $25 p/m (at IP40 only) you were going to do letters outside ? square ? you want them to change color all at the same time ? are you going to cover them with some sort of frost filter ? the cheap stuff should be fine. There are many shops and site to chose from, keep in mind if DHL is taking care of shipping you will be hit up for import fees.