Area & Volume

Surface Area of Solids

Finding the surface area of prisms, cylinders, pyramids, cones, and spheres.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Area Formulas

Unrolling a Curved Surface Flat

A cylinder's flat top and bottom are easy — they're just two circles. Before reading on, think about the curved side wrapping around the cylinder: if you could peel that curved surface off and lay it flat, what shape do you think it would become, and what would its dimensions be?

Unrolled flat, a cylinder's lateral (side) surface becomes an ordinary rectangle. Its height matches the cylinder's height exactly, and its width matches the distance around the circular base — the circumference, 2πr. That's why a cylinder's lateral surface area is simply height times circumference: 2πrh. Adding the two circular caps' areas, πr² each, gives the total surface area.

SA_{\text{cylinder}} = 2\pi r h + 2\pi r^{2}

A cone's lateral surface unrolls differently — not into a rectangle, but into a sector (a pie-slice) of a larger circle, since a cone tapers to a point rather than staying the same width all the way around. That sector's radius is the cone's slant height, ℓ (the distance along the cone's curved surface from the tip to the base edge, longer than the cone's straight-up height since it travels diagonally). The lateral surface area works out to πrℓ, and a pyramid's lateral surface area — the sum of its triangular faces — uses a directly analogous slant height in each triangle's height.

Worked Example — Finding a Cylinder's Surface Area

A cylinder has radius 4 and height 10. Find its total surface area. Lateral: 2π(4)(10) = 80π. Two caps: 2π(4²) = 32π. Total: 80π + 32π = 112π ≈ 351.9 square units.

Worked Example — Finding a Cone's Surface Area

A cone has radius 3 and slant height 7. Find its total surface area. Lateral: π(3)(7) = 21π. Base: π(3²) = 9π. Total: 21π + 9π = 30π ≈ 94.2 square units.

Tip

Never substitute a cone or pyramid's straight-up height for its slant height in a surface area formula — the two are only equal for a perfectly flat (zero-height) shape, and the slant height, found via the Pythagorean Theorem from the height and radius when needed, is always the longer of the two.

Common Mistakes

Using a cone's vertical height instead of its slant height when calculating lateral surface area.
Lateral surface area formulas for cones and pyramids specifically require the slant height — if only the vertical height is given, first find the slant height using the Pythagorean Theorem (it's the hypotenuse of a right triangle formed by the height and the radius).
Forgetting to add the base's area to a cylinder or cone's lateral surface area when the problem asks for total surface area rather than just lateral surface area.
Check carefully whether a problem wants lateral surface area (the curved side only) or total surface area (including the flat base or bases) before finalizing an answer.

Key Takeaways

A cylinder's lateral surface unrolls into a rectangle with height matching the cylinder and width matching its circumference.
A cone's lateral surface unrolls into a sector of a circle, using the slant height rather than the vertical height.
Slant height and vertical height are different measurements — mixing them up is one of the most common surface-area mistakes.

Summary

Unrolling curved surfaces into flat shapes explains where these surface area formulas come from. The final lesson in this unit moves from a solid's outer surface to the space it actually fills — its volume.

Track your progress

← Area Formulas Back to Area & Volume Volume of Solids→