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Math & Conversions Cone Volume Calculator Cone Volume Calculator ($\mathbf{V = \frac{1}{3}\pi r^2 h}$) Calculate the volume ($V$), base area ($B$), lateral area ($L$), and slant height ($l$) of a right circular cone. Input the radius ($r$) and the perpendicular height ($h$) below. Enter Dimensions Radius ($r$) Height ($h$, perpendicular) Calculate Properties Key Geometric Results Slant Height ($l$) Base Area ($B$) Lateral Area ($L$) Total Surface Area ($A$) Cone Volume ($V$) Step-by-Step Solution Formulas Used in Cone Geometry All calculations for a right circular cone derive from three fundamental geometric formulas: 1. Cone Volume ($V$) The volume of a cone is precisely one-third the volume of a cylinder with the same base and height. $$V = \frac{1}{3} B h = \frac{1}{3} \pi r^2 h$$ Where $r$ is the radius of the base and $h$ is the perpendicular height. 2. Surface Area Formulas The total surface area ($A$) is the sum of the circular base area ($B$) and the lateral (side) area ($L$). **Base Area ($B$):** Area of the circular base. $$\mathbf{B} = \pi r^2$$ **Lateral Area ($L$):** Area of the slanted side. $$\mathbf{L} = \pi r l$$ **Total Surface Area ($A$):** $$\mathbf{A} = \pi r^2 + \pi r l$$ 3. Slant Height ($l$) The slant height is necessary for surface area calculations. In a right cone, the radius ($r$), perpendicular height ($h$), and slant height ($l$) form a right triangle. Therefore, we use the Pythagorean theorem: $$l = \sqrt{r^2 + h^2}$$ Right Cone vs. Oblique Cone This calculator is designed for a **right cone**, where the apex (tip) is vertically aligned over the center of the base, making the height ($h$) perpendicular to the base radius ($r$). An **oblique cone** has its apex off-center, resulting in a slant height that is not uniform around the circumference. While the volume formula $V = \frac{1}{3} \pi r^2 h$ remains valid if the true perpendicular height ($h$) is known, calculating the surface area of an oblique cone is much more complex. Frequently Asked Questions (FAQ) What is the formula for the volume of a cone? The volume of a cone is one-third of the area of the base times the perpendicular height. The formula is $V = \frac{1}{3} \pi r^2 h$, where $r$ is the radius and $h$ is the height. What is the difference between height (h) and slant height (l)? The **height (h)** is the perpendicular distance from the apex (tip) of the cone to the center of the base. The **slant height (l)** is the distance from the apex to any point on the circumference of the base. In a right cone, they are related by the Pythagorean theorem: $r^2 + h^2 = l^2$. How do I find the base area of a cone? The base of a cone is a circle. Therefore, the base area (B) is calculated using the standard circle area formula: $B = \pi r^2$, where $r$ is the radius. What is the formula for the total surface area of a cone? The total surface area ($A$) is the sum of the base area ($B$) and the lateral surface area ($L$): $A = B + L$. This translates to the formula $A = \pi r^2 + \pi r l$, where $l$ is the slant height. Key Cone Formulas Volume $$V = \frac{1}{3} \pi r^2 h$$ Slant Height $$l = \sqrt{r^2 + h^2}$$ Total Surface Area $$A = \pi r (r + l)$$ Related Geometry Tools Cylinder Volume Calculator Sphere Volume Calculator Solid Geometry Calculator (All Shapes) Area Calculator (Base) Pythagorean Theorem Calculator
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