Geometry
Three-dimensional measurement
Surface area (prisms, pyramids, cones, cylinders, spheres)
20 practice questions
2 video lessons
Theory + worked examples
Surface Area of Solids
Common Core Geometry • Standard G-MG.1 • Three-Dimensional Measurement
Surface Area of Solids is the opening topic of Three-Dimensional Measurement in the Common Core State Standards. It is aligned to Standard G-MG.1, which requires students to use geometric shapes and their measures to model objects, including their surface area.
Surface area is the total area of every face and curved surface of a solid.
Theory
Surface area is the total area of every face and curved surface of a solid — the area you would paint or wrap. Key formulas:
- 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: \(4\pi r^2\).
A net helps: unfolding the solid flat shows every face to add up.
Surface area is the total outer area of a solid.
Surface-area formulas for the round solids.
The round-solid surface areas:
\[\text{cylinder } 2\pi r^2+2\pi rh,\quad \text{cone } \pi r^2+\pi r\ell,\quad \text{sphere } 4\pi r^2\]
The cone uses the slant height \(\ell\), not the vertical height.
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.
Example 1 — Cylinder surface area
Find the surface area of a cylinder with radius \(3\) and height \(5\) (leave \(\pi\)).
Solution
Two circular bases plus the curved side.
| \(SA\) | \(=\) | \(2\pi r^2+2\pi rh\) |
| \(=\) | \(2\pi(9)+2\pi(3)(5)\) | |
| \(=\) | \(18\pi+30\pi=48\pi\) |
Example 2 — Sphere surface area
Find the surface area of a sphere of radius \(6\) (leave \(\pi\)).
Solution
Use \(SA=4\pi r^2\).
| \(SA\) | \(=\) | \(4\pi(6)^2=144\pi\) |
Example 3 — Cone surface area
Find the surface area of a cone with radius \(4\) and slant height \(9\) (leave \(\pi\)).
Solution
Base plus lateral surface: \(\pi r^2+\pi r\ell\).
| \(SA\) | \(=\) | \(\pi(16)+\pi(4)(9)\) |
| \(=\) | \(16\pi+36\pi=52\pi\) |
Example 4 — Prism surface area
A rectangular prism is \(4\times 3\times 2\). Find its surface area.
Solution
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
A cone's lateral area uses the slant height \(\ell\), not the vertical height.
Include both bases for a cylinder or prism's total surface area.
Surface area is in square units; volume is cubic.
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.
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