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Radiometric dating uses the known, constant decay rates of radioactive isotopes to calculate the age of minerals and rocks by measuring the ratio of parent to daughter isotopes. Different isotope systems are suited to different time ranges and materials: carbon-14 (half-life ~5,730 years) dates organic material up to ~50,000 years old; potassium-40/argon-40 dates volcanic minerals millions to billions of years old; uranium-lead dating of zircon crystals can yield ages close to 4 billion years. The method requires a closed system assumption—that no parent or daughter isotopes were added or lost after mineral crystallization—which is tested using concordia diagrams (for U-Pb) or isochron plots. Radiometric dating calibrates the geological time scale and has provided overwhelming evidence that Earth is 4.54 billion years old.
Working through a decay calculation (given a measured parent/daughter ratio and a known half-life, solve for age) connects the physics of radioactive decay directly to the geological application. Understanding why carbon-14 is useless for dating dinosaur bones (too old by ~60 million years, essentially no C-14 remains) reinforces the importance of choosing the appropriate isotope system for the time range of interest.
You already know from your prerequisites that radioactive isotopes decay at a constant, predictable rate described by the half-life: the time it takes for exactly half of a sample to decay from the parent isotope to a stable daughter product. Radiometric dating turns this into a clock. If you measure the ratio of parent to daughter atoms in a mineral, and you know the half-life, you can calculate how long ago that mineral formed — because the only way daughter atoms are present is through radioactive decay of the parent since the mineral crystallized.
The key starting condition is the crystallization event. When a mineral like zircon or feldspar forms from molten rock, it incorporates certain elements based on its crystal chemistry — for example, zircon accepts uranium but strongly rejects lead. This means that at time zero, the mineral contains essentially pure parent isotope (uranium) and no daughter (lead). From that moment, uranium decays to lead at a known rate. When a geologist measures the U/Pb ratio today, the amount of lead is the accumulated "clock reading" since crystallization.
Different decay systems are calibrated for different time ranges. Carbon-14 (half-life ~5,730 years) dates organic materials up to ~50,000 years old — it works because living organisms continuously exchange carbon with the atmosphere, fixing carbon-14 until death, after which no new C-14 enters and the existing C-14 decays. After ~10 half-lives, the remaining C-14 is below detection limits, which is why carbon dating is useless for dinosaur bones (66+ million years old). For ancient rocks, geologists use potassium-40/argon-40 (half-life ~1.25 billion years) or uranium-238/lead-206 (half-life ~4.47 billion years), which have yielded consistent ages for the oldest terrestrial zircons at ~4.4 billion years.
The method's validity rests on the closed system assumption: no parent or daughter isotopes entered or left the mineral after crystallization. Geologists test this using concordia diagrams (for U-Pb systems) or isochron plots, which reveal whether a sample has remained closed or has been disturbed by later heating or fluid activity. When multiple dating methods agree on the same age for the same rock — called concordance — this provides strong validation of both the method and the closed system assumption. The consistent picture from thousands of such measurements across all continents is that Earth is 4.54 billion years old.