New to exercising, is this plan ok?

I know this question is probably getting old but I've only read answers that partially answer my question. So for someone who is well-versed in fitness, am I on the right path? I want to look good, which basically means have defined muscles that are moderately large, which I expect means I need lower my body fat percentage. I'm a 17 year old male @ 175lbs and 19% body fat (but I'm not sure about the body fat because I can't find the suprailiac to save my life).

The plan I devised from what I've read is this...

5 days a week, I run on an elliptical machine for 50-60 minutes straight and I keep my heart rate 170-185bpm. Some of that time is spent doing HIIT and the rest is just constant pace.
Every other day is muscle building. For arms and upper body, I do pushups until failure, pull-ups until failure, chin-ups until failure, bicep curls (5-7 reps until failure per set. I raise weight to keep the reps the same and it's 32.5lbs currently), I use the same bicep weight for tricep dips behind your head until failure, I use lighter weights (15lbs) for straight arm extensions to my side and front until failure. I do 5 sets of all of those (all until failure) every other day.
For core, I do a long combination of leg raises, normal crunches, cross-arm crunches, alternating crunches, one-side crunches, and maybe a few other things I'm forgetting until it burns and I can't hardly do them correctly any more.
I don't do many leg exercises because I don't want them big but rather toned which I hope will come as my body fat % lowers.

My diet stays within 1800-2000cal per day. I used a calculator to find my BMR was 1800cal so that 1800 plus whatever I exercise (usually 500-700cal on elliptical days). I eat 1800-2000 regardless of if I burn extra calories or not. My diet usually consists of 2 servings of skim milk and a scoop of whey protein 3 times a day which is already 900cal which sucks because that's only like 1000cal for solid food. But the rest of that usually goes to Special K protein meal bars and turkey/cheese sandwhiches. I'd love to eat this diet every day forever but it seems to be high in sodium so I gotta switch it up every now and then. I have the occasional apple, pieces of toast with ICBINB spray, egg and cheese sandwich, apples, yoplait fat free yogurt. It seems to be pretty decent because it's 1800-2000cal a day and the protein/carb/fat spread is like 30p/40c/20f. Any tips or corrections or anything for feedback? My goal is to look decent by the summer so I still have 4 months. If not then by college which is 7 months. I know that's not a long time but I just hope I can get somewhere

I guess I need to understand what resistance training and cardio do for you. I mean resistance training I see primarily to gain muscle mass as long as you have good amounts of protein in your diet. And I see cardio just as a way to burn extra calories so that your diet doesn't put those extra calories on as fat. Do they really have other major functions? I mean if you do them together, won't you lose fat from cardio and build muscle from resistance? I know this is incredibly simplistic and probably wrong lol but please explain.
 
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