Tapered drinkware tumblers and travel mugs are geometric conical frustums rather than straight cylinders. Applying a standard rectangular graphic around a tapered tumbler forces the seam out of alignment, creating diagonal seam mismatches and top-to-bottom edge distortion. This tool calculates the exact circular arc radius and sweep angle required to distort rectangular designs into a tapered fan template for seamless sublimation wrapping.

### Geometric Conical Frustum Formulas

When an unrolled cone surface (frustum) is laid flat on paper, the top and bottom circumferences form concentric circular arcs: 1. **Circumference-to-Radius Ratios:** $$r_1 = \frac{\text{Top Circumference}}{2\pi}, \quad r_2 = \frac{\text{Bottom Circumference}}{2\pi}$$ 2. **Slant Height ($L$):** $$L = \sqrt{\text{Height}^2 + (r_1 - r_2)^2}$$ 3. **Inner Arc Radius ($R_1$) & Outer Arc Radius ($R_2$):** $$R_1 = \frac{r_1 \times L}{r_1 - r_2}, \quad R_2 = R_1 - L$$ 4. **Arc Sweep Angle (\theta):** $$\theta = \left(\frac{r_1}{R_1}\right) \times 360^\circ = \left(\frac{\text{Top Circumference}}{R_1}\right) \times \left(\frac{180^\circ}{\pi}\right)$$

### Worked Conical Arc Example

Using a standard 20oz tapered skinny tumbler preset: - **Top Circumference:** 9.25 inches - **Bottom Circumference:** 8.50 inches - **Slant Vertical Height:** 8.20 inches - **Circumference Difference:** $9.25 - 8.50 = 0.75$ inches - **Radius Delta:** $\Delta r = 0.75 / (2\pi) \approx 0.119$ inches - **Calculated Slant Height:** $L = \sqrt{8.20^2 + 0.119^2} \approx 8.201$ inches - **Outer Unroll Arc Radius:** $R_1 \approx 101.1$ inches - **Sweep Angle:** $\theta \approx 32.8^\circ$ - The printed transfer must be warped into an arc of $32.8^\circ$ so when heat-pressed onto the metal tumbler, the graphic edges meet along a perfectly vertical seam.

### In-Browser Processing Guarantee

All conical arc trigonometric geometry and image warp rendering calculate directly inside your web browser using HTML5 Canvas. Artwork files stay strictly on your local computer.