2 semantic clarification
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Michael's answer covers well how the area and dimensions of the two sensor sizes relate, but does not conceptually explain how or why that appears to affects focal length. I'll try to explain that.

A lens designed for full frame 35mm (FF35) is designed to project a circular image large enough to cover a 36x24mm sensor / film frame. So what we have to imagine is: what would happen if we used a smaller sensor that only recorded a portion of that circle?

Get a camera with a 36x24mm sensor. Make Image A with a 200mm lens, and then Image B with a 300mm lens. Image A has a 12.3 degree diagonal field of view, Image B 8.2 degree.

Now, open Image A in Photoshop and crop it down to 2/3rds (1/1.5) its original size. It will now have the same 8.2 degree field of view as Image B. You've cropped your image so that it has a field of view equivalent to a lens with a focal length 1.5 times longer. Well, that's exactly what happens when you put your FF35-format 200mm lens on a camera with an APS-C (24x16, 2/3rd the size of FF35) sensor! So APS-C can be said to have a "1.5x crop factor" compared to FF35.

Michael's answer covers well how the area and dimensions of the two sensor sizes relate, but does not conceptually explain how or why that affects focal length. I'll try to explain that.

A lens designed for full frame 35mm (FF35) is designed to project a circular image large enough to cover a 36x24mm sensor / film frame. So what we have to imagine is: what would happen if we used a smaller sensor that only recorded a portion of that circle?

Get a camera with a 36x24mm sensor. Make Image A with a 200mm lens, and then Image B with a 300mm lens. Image A has a 12.3 degree diagonal field of view, Image B 8.2 degree.

Now, open Image A in Photoshop and crop it down to 2/3rds (1/1.5) its original size. It will now have the same 8.2 degree field of view as Image B. You've cropped your image so that it has a field of view equivalent to a lens with a focal length 1.5 times longer. Well, that's exactly what happens when you put your FF35-format 200mm lens on a camera with an APS-C (24x16, 2/3rd the size of FF35) sensor! So APS-C can be said to have a "1.5x crop factor" compared to FF35.

Michael's answer covers well how the area and dimensions of the two sensor sizes relate, but does not conceptually explain how or why that appears to affects focal length. I'll try to explain that.

A lens designed for full frame 35mm (FF35) is designed to project a circular image large enough to cover a 36x24mm sensor / film frame. So what we have to imagine is: what would happen if we used a smaller sensor that only recorded a portion of that circle?

Get a camera with a 36x24mm sensor. Make Image A with a 200mm lens, and then Image B with a 300mm lens. Image A has a 12.3 degree diagonal field of view, Image B 8.2 degree.

Now, open Image A in Photoshop and crop it down to 2/3rds (1/1.5) its original size. It will now have the same 8.2 degree field of view as Image B. You've cropped your image so that it has a field of view equivalent to a lens with a focal length 1.5 times longer. Well, that's exactly what happens when you put your FF35-format 200mm lens on a camera with an APS-C (24x16, 2/3rd the size of FF35) sensor! So APS-C can be said to have a "1.5x crop factor" compared to FF35.

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Michael's answer covers well how the area and dimensions of the two sensor sizes relate, but does not conceptually explain how or why that affects focal length. I'll try to explain that.

A lens designed for full frame 35mm (FF35) is designed to project a circular image large enough to cover a 36x24mm sensor / film frame. So what we have to imagine is: what would happen if we used a smaller sensor that only recorded a portion of that circle?

Get a camera with a 36x24mm sensor. Make Image A with a 200mm lens, and then Image B with a 300mm lens. Image A has a 12.3 degree diagonal field of view, Image B 8.2 degree.

Now, open Image A in Photoshop and crop it down to 2/3rds (1/1.5) its original size. It will now have the same 8.2 degree field of view as Image B. You've cropped your image so that it has a field of view equivalent to a lens with a focal length 1.5 times longer. Well, that's exactly what happens when you put your FF35-format 200mm lens on a camera with an APS-C (24x16, 2/3rd the size of FF35) sensor! So APS-C can be said to have a "1.5x crop factor" compared to FF35.