1. The problem is to explain and understand the geometry theorem 3.14, which is commonly known as the value of $\pi$ in geometry, relating the circumference of a circle to its diameter. 2. The theorem states that the ratio of the circumference $C$ of any circle to its diameter $d$ is a constant, denoted by $\pi$: $$\pi = \frac{C}{d}$$ 3. Important rules: - The diameter $d$ is twice the radius $r$, so $d = 2r$. - The circumference formula can also be written as $C = 2\pi r$. 4. To understand this, consider a circle with radius $r$. 5. Using the formula for circumference: $$C = 2\pi r$$ 6. Dividing both sides by the diameter $d = 2r$: $$\frac{C}{d} = \frac{2\pi r}{2r}$$ 7. Canceling $2r$ in numerator and denominator: $$\frac{C}{\cancel{2r}} = \frac{2\pi r}{\cancel{2r}} = \pi$$ 8. This shows that the ratio $\frac{C}{d}$ is always $\pi$, approximately 3.14159. 9. This theorem is fundamental in geometry and is used to calculate lengths and areas involving circles. Final answer: The geometry theorem 3.14 states that the ratio of the circumference of a circle to its diameter is $\pi$.