Surface area (prisms, pyramids, cones, cylinders, spheres)
Surface Area of Solids
Surface Area of Solids is the opening topic of Three-Dimensional Measurement in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.11(C), which requires students to apply the formulas for the total and lateral surface area of three-dimensional figures to solve problems.
Surface area is the total area of every face and curved surface of a solid.
Theory
- Prism / box: sum of all face areas.
- Cylinder: \(2\pi r^2+2\pi rh\) (two bases + curved side).
- Cone: \(\pi r^2+\pi r\ell\) (base + lateral, slant \(\ell\)).
- Sphere: \(4\pi r^2\).
The round-solid surface areas:
How to find surface area
- Identify the solid and its faces/surfaces.
- Add the area of each face (or use the formula).
- Use the slant height for a cone's lateral area.
- Keep \(\pi\) exact for round solids.
Two circular bases plus the curved side.
| \(SA\) | \(=\) | \(2\pi r^2+2\pi rh\) |
| \(=\) | \(2\pi(9)+2\pi(3)(5)\) | |
| \(=\) | \(18\pi+30\pi=48\pi\) |
Use \(SA=4\pi r^2\).
| \(SA\) | \(=\) | \(4\pi(6)^2=144\pi\) |
Base plus lateral surface: \(\pi r^2+\pi r\ell\).
| \(SA\) | \(=\) | \(\pi(16)+\pi(4)(9)\) |
| \(=\) | \(16\pi+36\pi=52\pi\) |
Add the areas of the three pairs of faces.
| \(SA\) | \(=\) | \(2(4\cdot 3)+2(4\cdot 2)+2(3\cdot 2)\) |
| \(=\) | \(24+16+12=52\) |
Common pitfalls
Frequently asked questions
What is surface area?
The total area of all the faces and curved surfaces of a 3D solid.
What is the surface area of a cylinder?
\(2\pi r^2+2\pi rh\): two circular bases plus the curved side.
What is the surface area of a sphere?
\(4\pi r^2\).
Which height does a cone's surface area use?
The slant height \(\ell\) for the lateral surface, not the vertical height.