Math

Surface Area

2 min readEasy5-question drill

Surface area shows up on the test wrapped around boxes, cans, and prisms — and if you know the simple trick of adding up faces, these become free points.

Where this lesson fits

Surface Area

You are here

→Area and Volume →Circles

Surface area is the total area of all the outside surfaces of a 3D shape. Think of it as how much wrapping paper you'd need to cover a box completely, or how much paint to coat the outside. The units are always square units (like cm² or in²) — never cubic, because area measures flat covering, not the space inside (that's volume).

The big idea: a 3D shape is made of flat (or curved) faces. To get surface area, find the area of each face and add them all up.

Rectangular prism (a box). A box has 6 faces that come in 3 matching pairs. If the box has length l, width w, and height h, the formula is:

SA = 2lw + 2lh + 2wh

That's just: (top + bottom) + (front + back) + (two sides). Each pair is identical, which is why everything is multiplied by 2.

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A box has 6 faces in 3 matching pairs: top/bottom, front/back, two sides.

Cube. A cube is a box where all edges are equal length s. It has 6 identical square faces, each with area s², so:

SA = 6s²

Cylinder (a can). A cylinder has two circular ends plus the curved side. If you peel off the curved side and unroll it, it becomes a rectangle. The formula:

SA = 2πr² + 2πrh

Here r is the radius and h is the height. The 2πr² is the two circles (top and bottom), and 2πrh is the unrolled rectangle — its width is the circle's circumference (2πr) and its height is h.

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A cylinder's surface = two circular ends plus the curved side, which unrolls into a rectangle.

The key strategy on the test: Most surface-area questions are really just careful arithmetic. Identify the shape, write the formula, plug in the numbers, and don't mix up radius and diameter. Sometimes the test gives you a 2D net (an unfolded shape) — then literally add up the rectangles you see.

A related basic skill the test loves: area of flat shapes. Rectangle area = length × width. Triangle area = ½ × base × height. These feed directly into surface-area problems, since faces are usually rectangles or triangles.

Know the three formulas above and you'll handle almost every surface-area question you see.

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Quick check

A rectangle has length 3 and width 14. What is its area?

Your answer:

Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).

Worked examples

Example 1

A rectangular box has length 5, width 3, and height 2. What is its surface area?

Example 2

A cube has a surface area of 54 square inches. What is the length of one edge, in inches?

Example 3

A closed cylindrical can has radius 3 and height 7. What is its total surface area? (Use π and give your answer in terms of π.)

Common pitfalls

Confusing surface area with volume

Surface area uses square units and adds up faces; volume uses cubic units and multiplies dimensions. If a cube has edge 3, surface area is 6(3²)=54 but volume is 3³=27. Read whether the question asks to cover (area) or fill (volume).

Using diameter instead of radius

Cylinder formulas use the radius. If a problem gives the diameter, cut it in half first. Plugging diameter straight into 2πr² doubles your radius and wrecks the answer.

Forgetting a pair of faces

A box has 6 faces in 3 pairs — every term in 2lw + 2lh + 2wh has a 2. Dropping a 2 or skipping the top/bottom is the most common arithmetic slip.

Forgetting the can's two ends

For a closed cylinder you need both circles plus the side. If the problem says 'open' or 'tube,' you may drop one or both circles — read the wording.

Key takeaways

Surface area = total area of all outside faces; answer in square units.
Rectangular prism: SA = 2lw + 2lh + 2wh; cube: SA = 6s².
Cylinder: SA = 2πr² + 2πrh (two circles + unrolled rectangle).
Always check radius vs. diameter and don't drop any face.
You can solve backward: given surface area, plug in and solve for a missing dimension.

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Try it yourself

5 practice questions on Surface Area, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Surface Area

You are here

→Area and Volume →Circles