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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
Take one third for pointy solids, earn slant height from Pythagoras, and drop hidden faces from composite area.
More to read
3 Ways to Find the Surface Area of Cones
A cone net is one circle plus one curved sector, and that picture drives both formulas. The slant height runs up the slope and comes from Pythagoras on radius and perpendicular height. Curved area is pi times radius times slant height, and adding the base circle gives the total. Never plug the slant height into the volume formula by mistake. Volume wants the perpendicular height, always the shortest drop from tip to base.
Volume formulas review
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. Spheres follow their own rule: four thirds pi r cubed, which is about four times the matching cone. Composite solids just add pieces, then subtract any faces that end up hidden inside. Learn the family once and every solid becomes a remix.
9.6 Solve Geometry Applications: Volume and Surface Area
Real jobs constantly ask for volumes and surface areas of containers, roofs, and tanks. The method never changes: name the solid, write its formula, substitute with matching units, and sanity check the size. Cones and pyramids shrink their prism cousins by a factor of one third, so expect smaller numbers than the box around them. For surface area, unfold mentally and count every exposed face. An answer ten times too big usually means a radius doubled into a diameter somewhere.
7.3.6: Applying Volume and Surface Area
Applying volume means choosing the right tool before calculating anything. Prisms and cylinders multiply base by length, pyramids and cones take one third of that, and spheres use four thirds pi r cubed. Draw the solid and label which height you have: perpendicular or slant. Composite shapes split into familiar parts, like a hemisphere glued onto a cylinder. Add the volumes, then pause on surface area, because joined faces no longer need paint.
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.