--- title: "Volume and Surface Area of Cones, Spheres and Pyramids" description: "Apply the standard formulae for pyramids, cones and spheres, and find a slant height when only the perpendicular height is given." canonical: https://lightmysky.com/learn/mathematics/volume-and-surface-area-of-cones-spheres-and-pyramids-mt_s4K5YMBSzG source: https://lightmysky.com/learn/mathematics/volume-and-surface-area-of-cones-spheres-and-pyramids-mt_s4K5YMBSzG.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Volume and Surface Area of Cones, Spheres and Pyramids Apply the standard formulae for pyramids, cones and spheres, and find a slant height when only the perpendicular height is given. Subject: Mathematics · Area: Geometry · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/volume-and-surface-area-of-cones-spheres-and-pyramids-mt_s4K5YMBSzG ## Ready when they can - Use the one-third factor for a pyramid or cone and say which height the formula wants - Find a cone's slant height from its radius and perpendicular height before computing curved surface area - Handle a composite solid such as a hemisphere on a cylinder by adding volumes and adjusting which faces are exposed ## Lesson: Pointy solids and their measures Pointy solids hold exactly one third of their boxy neighbours. A pyramid fills one third of the prism with the same base and height, and a cone fills one third of the matching cylinder. Volume always wants the perpendicular height, the shortest drop from tip to base, never the slant height. **Example.** Take a cone with radius 5 cm and perpendicular height 12 cm. Radius, height, and slant form a right triangle, so slant squared is 25 plus 144, which is 169, giving 13 cm. Curved area is pi times radius times slant, and the total adds the base circle. Spheres follow their own rule: four thirds pi r cubed, about four times the matching cone. Composite solids split into familiar parts. Add the volumes, then pause on surface area, because joined faces need no paint: a hemisphere on a cylinder buries the flat face and the top face. **Tip.** Work every solid the same way: name it, write its formula, substitute with matching units, and sanity check the size. Expect less than the box around it. If an answer looks wild, check you never doubled a radius into a diameter. **Recap.** Take one third for pointy solids, earn slant height from Pythagoras, and drop hidden faces from composite area. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Pythagoras' Theorem](https://lightmysky.com/learn/mathematics/pythagoras-theorem-mt_1VmTUxBrNd) - [Volume and Surface Area of Prisms and Cylinders](https://lightmysky.com/learn/mathematics/volume-and-surface-area-of-prisms-and-cylinders-mt_JTLJ5LDrYh) ## Opens up - [Volumes of Revolution: Disks, Washers and Shells](https://lightmysky.com/learn/mathematics/volumes-of-revolution-disks-washers-and-shells-mt_mzZ22MkWD4)