Subjects geometry

Surface Volume Formulas

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Surface Volume Formulas


1. The problem asks to find formulas for points 8, 9, 10 related to cones, spheres, and hemispheres. 2. Based on Chapter 11.5 summary, points 1-7 are about surface areas and volumes. 3. We can reasonably assume points 8, 9, 10 involve important related formulas not listed, commonly: - 8. Slant height relation in cone: $l = \sqrt{r^2 + h^2}$, where $r$ is the base radius, $h$ is height, and $l$ is slant height. - 9. Volume of hemisphere: Half the volume of a sphere, so $$V_{hemisphere} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3$$ - 10. Surface area of right circular cone including base: Total area $= \pi r (l + r)$ (already point 2). Another useful formula could be the lateral surface area of frustum, or you may want to consider: Curved surface area of frustum = $\pi (r_1 + r_2) l$, where $r_1$ and $r_2$ are radii of the two circular ends, $l$ is slant height. 4. Explanation: - For 8, slant height $l$ relates the radius and height by the Pythagorean theorem. - For 9, volume of hemisphere is half the sphere's volume. - For 10, the curved surface area of a conical frustum formula is another important extension. Final answers: 8. $l = \sqrt{r^2 + h^2}$ 9. $V = \frac{2}{3} \pi r^3$ 10. Curved Surface Area of frustum $= \pi (r_1 + r_2) l$ These complete the summary with important related formulas.