What is the surface area of a solid?

It is the sum of the areas of all its faces — the area of its net, that is, how much paper or sheet metal it takes to cover it. It is measured in square units: cm², m².

How do you find the surface area of a cuboid?

The faces come in identical pairs, so A = 2(ab + bc + ac). For edges of 4 cm, 3 cm and 2 cm that gives A = 2(12 + 6 + 8) = 52 cm².

How do you find the surface area of a cylinder?

Add the two bases (circles of area πr² each) to the lateral surface, which unrolls into a rectangle with sides 2πr and h. Together A = 2πr² + 2πrh = 2πr(r + h). For r = 3 cm and h = 5 cm that is 48π ≈ 150.72 cm².

How do you find the surface area of a sphere?

From A = 4πr² — exactly four areas of a circle of the same radius. For r = 3 cm we get A = 4π · 9 = 36π ≈ 113.04 cm².

What is the slant height of a cone?

It is the segment joining the apex to a point on the rim of the base, written l. Together with the radius and the height it forms a right triangle, so l = √(r² + h²). Surface area takes the slant height; volume takes the height.

How does the slant height of a pyramid differ from its height?

The height H runs from the apex perpendicular to the base, inside the solid. The slant height l lies on a lateral face and runs from the apex to the midpoint of a base edge. It is the longer of the two, and it is the one the surface-area formula takes: l = √(H² + (a/2)²).

What units is surface area given in?

Square ones — cm², m² — because it is still an area, just computed for the skin of a solid. The cubic unit (cm³) belongs to volume, which is an entirely different quantity.

How much does surface area grow when every dimension is doubled?

Four times, because 2² = 4 — an area depends on two dimensions. Volume, meanwhile, grows eight times. That is why large animals lose heat more slowly than small ones: the surface grows more slowly than the mass.

Intermediate level

Surface area of solids

Surface area is the area of a solid’s net — how much material it takes to cover it. See the formulas for a cuboid, a cylinder, a pyramid, a cone and a sphere, and why a cone and a pyramid are measured by the slant height rather than the height.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

Volume of solidsVolume says how much fits inside a solid. See the formulas for a cuboid, a cylinder, a pyramid, a cone and a sphere, why a third appears in the pyramid and the cone, and where the cubic unit comes from.
The Pythagorean theoremIn a right triangle a² + b² = c². See why it works, how to find the hypotenuse and a leg, what Pythagorean triples are, and how the converse lets you check whether a corner really is square.

All formulas

The general rule
$A = A_b + A_l$
base area plus lateral area
Cuboid
$A = 2(ab + bc + ac)$
three pairs of identical faces
Cube
$A = 6a^2$
six identical squares
Cylinder
$A = 2\pi r^2 + 2\pi rh = 2\pi r(r + h)$
two circles and a rectangle with sides 2πr and h
Pyramid on a square base
$A = a^2 + 2al$
l is the slant height of a face, not the height of the solid
Cone
$A = \pi r^2 + \pi rl = \pi r(r + l)$
l is the slant height — from the apex to the rim of the base
Slant height of a cone
$l = \sqrt{r^2 + h^2}$
from the Pythagorean theorem for the radius and the height
Sphere
$A = 4\pi r^2$
exactly four areas of a great circle

Volume says how much fits inside a solid. Surface area says how much material it takes to cover it — which is why we are back to square units.

Surface area is the area of the net

Cut a box along its edges and lay it flat. What you are left with is the net of the solid: a handful of ordinary plane figures whose areas you already know how to find. The surface area is their sum.

A cuboid has six faces but only three different ones: the front a × c, the side b × c and the top a × b. Each appears twice.

The general rule is always the same:

A = A_b + A_l

where $A_b$ is the base area (or the area of both bases) and $A_l$ is the lateral area.

Cuboid and cube

The faces come in identical pairs, so it is enough to work out three of them and double the result:

A = 2(ab + bc + ac)

Find the surface area of a cuboid with edges of 4 cm, 3 cm and 2 cm.

In a cube every face is the same, so the formula collapses to:

A = 6a^2

For an edge of $3\,\text{cm}$ that gives $6 \cdot 9 = 54\,\text{cm}^2$ .

Cylinder

The net of a cylinder is two circles and one rectangle. The rectangle comes from unrolling the lateral surface, so its longer side is the circumference of the base, $2\pi r$ , and its shorter side is the height $h$ .

A cylinder: two circles of radius r plus a lateral surface which, cut open, is a 2πr by h rectangle.

A = \underbrace{2\pi r^2}_{\text{two bases}} + \underbrace{2\pi rh}_{\text{lateral area}} = 2\pi r(r + h)

Find the surface area of a cylinder with a radius of 3 cm and a height of 5 cm. Take π ≈ 3.14.

Pyramid: the slant height, not the height

The net of a pyramid on a square base is a square and four triangles. The base of each triangle is the edge $a$ , and its height is the segment $l$ — the slant height of the face.

The height H runs inside the solid, perpendicular to the base. The slant height lies on a lateral face and is longer than H — and it is the one the area formula takes.

One triangle has area $\frac{1}{2}al$ , and there are four of them:

A = a^2 + 4 \cdot \frac{1}{2}al = a^2 + 2al

The slant height follows from the Pythagorean theorem — it is the hypotenuse of a right triangle with legs $H$ and $\frac{a}{2}$ :

l = \sqrt{H^2 + \left(\frac{a}{2}\right)^2}

A pyramid has a square base of side 6 cm and a height of 4 cm. Find its surface area.

Cone: the slant height again

In a cone the part played by the pyramid's slant height is played by the slant height $l$ — the segment from the apex to the rim of the base. Unrolled, the lateral surface is a sector of a circle of radius $l$ , and its area is $\pi rl$ .

The radius r, the height h and the slant height l form a right triangle, so l = √(r² + h²).

A = \pi r^2 + \pi rl = \pi r(r + l)

The slant height is almost never given alongside the height — it has to be worked out:

l = \sqrt{r^2 + h^2}

A cone has a base radius of 3 cm and a height of 4 cm. Find its surface area. Take π ≈ 3.14.

Sphere

A sphere cannot be laid out flat — every map of the world is proof of that. Even so its surface area is surprisingly simple: exactly four areas of a circle of the same radius.

A = 4\pi r^2

A sphere of radius r. Its surface area is four areas of a great circle — a circle of that same radius r.

Find the surface area of a sphere of radius 3 cm. Take π ≈ 3.14.

Units

Surface area is still an area, so it is measured in $\text{cm}^2$ , $\text{m}^2$ and hectares — not in $\text{cm}^3$ . When converting units of area the length factor is squared: $1\,\text{m}^2 = 10{,}000\,\text{cm}^2$ . For actual values, our area converter will do the arithmetic.

Exercises

If the question does not name a unit, type it together with the answer, e.g. 52 cm². If the unit is given in brackets (because $\pi$ shows up in the answer), type the number alone, rounded to two decimal places. Slant heights are always given in the question — you never have to derive one here.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

Surface area of a cuboid: a = 4 cm, b = 6 cm, c = 6 cm

Common mistakes

The height instead of the slant height — surface area takes the segment lying on a lateral face, which is longer than the height.
Forgetting the cylinder's second base — there are two circles, hence $2\pi r^2$ .
Counting a base that is not there — a sphere has no base, and a cone has only one.
A cubic unit — $52\,\text{cm}^2$ , not $52\,\text{cm}^3$ ; the cube belongs to volume.
Mixing up the sphere's formulas — $4\pi r^2$ is the area, $\frac{4}{3}\pi r^3$ is the volume; the area has the radius squared.
Taking the lateral area for the whole — $A_l$ is only a part; the general rule is $A = A_b + A_l$ .

Formula card

Topic: Surface area of solids

The general rule
$A = A_b + A_l$
base area plus lateral area
Cuboid
$A = 2(ab + bc + ac)$
three pairs of identical faces
Cube
$A = 6a^2$
six identical squares
Cylinder
$A = 2\pi r^2 + 2\pi rh = 2\pi r(r + h)$
two circles and a rectangle with sides 2πr and h
Pyramid on a square base
$A = a^2 + 2al$
l is the slant height of a face, not the height of the solid
Cone
$A = \pi r^2 + \pi rl = \pi r(r + l)$
l is the slant height — from the apex to the rim of the base
Slant height of a cone
$l = \sqrt{r^2 + h^2}$
from the Pythagorean theorem for the radius and the height
Sphere
$A = 4\pi r^2$
exactly four areas of a great circle

A cuboid with edges a, b, c. It has six faces, but they come in identical pairs — hence A = 2(ab + bc + ac).

A cone: base radius r, height h and slant height l. Surface area takes the slant height: A = πr(r + l).

Frequently asked questions

A luminous right triangle with squares built on its sides drawn as a constellation, beside a circle with its radius, against a cosmic nebula in violet and magenta

Geometry

Why we measure area in squares

Perimeter is counted in metres, area in square metres — and that single exponent is behind most of the mistakes people make in plane geometry. It is why 1 m² is not 100 cm², why two plots with identical area can need wildly different amounts of fencing, and why the Pythagorean theorem is really a statement about the areas of squares rather than the lengths of sides.

11 min