What is the difference between surface area and volume?

Surface area is the total area of the outside faces — the paint or wrapping needed. Volume is the space inside — what the solid would hold. Surface area is in square units, volume in cubic units.

When do I use slant height and when do I use vertical height?

Cone volume uses the vertical height h; cone surface area uses the slant height l. They are linked by l = \sqrt{r^2 + h^2}, so if a problem gives you one you can always find the other.

Why do cones and pyramids have a one-third in the formula?

Because a cone fits exactly three times inside the cylinder with the same base and height — and a pyramid three times inside its matching prism. It is a genuine geometric fact, provable with calculus, not an arbitrary constant.

How do I find the volume of a shape that is not in the formula list?

If it has a constant cross-section, find that cross-section’s area and multiply by the length — that covers every prism, whatever the shape of the end. If it is a composite, split it into standard solids and add the volumes.

What happens to volume when I double the dimensions?

Volume multiplies by 2^3 = 8, while surface area multiplies by 2^2 = 4. This scale-factor relationship is a favourite exam question and follows directly from the powers in the formulas.

Can Photo Math Solver read a 3D diagram or a net?

Yes — screenshot the solid with its labels, and the solver identifies the shape, sorts out which measurement is the radius, height, or slant, and shows the substitution.

You can start solving surface area and volume problems for free by adding the Chrome extension — try it on your own worksheet before deciding if you need anything more.

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Photo Math Solver Solved

\text{Cuboid: } 4 \times 5 \times 6

1Volume: multiply all three dimensions, $V = 4 \cdot 5 \cdot 6 = 120$ cubic units.
2Surface area: three pairs of faces, $2(4\cdot5) + 2(4\cdot6) + 2(5\cdot6)$ .
3 $SA = 40 + 48 + 60 = 148$ square units.

On this page01Definition 02Formulas 03Methods 04Examples 05Practice 06Mistakes 07FAQ

01Definition

What are surface area and volume?

Volume measures how much space a solid contains — how much water it would hold. Surface area measures the total area of its outside faces — how much wrapping paper or paint it would take. One is about the inside, the other about the skin.

The units tell you which is which. Volume multiplies three lengths, so it comes out in cm³ or m³. Surface area is still a collection of flat areas added up, so it stays in cm² or m². A volume answer without a cubed unit is a warning sign.

Most volume formulas share one idea: cross-section times height. A prism or cylinder is a shape swept straight through space, so its volume is just base area × height. Cones and pyramids taper to a point and hold exactly one third as much as the prism or cylinder that boxes them in.

Where you'll actually use this

Packaging design: minimising surface area (cardboard cost) for a required volume.
Tank and pool capacity in litres, converted straight from a volume in cm³ or m³.
Paint and insulation coverage, which depends on surface area rather than volume.
Shipping and freight, where cost is charged by volumetric weight of the box.
Cooking and scaling recipes: doubling every pan dimension multiplies capacity by eight, not two.

Prism: A solid with the same cross-section all the way along its length.
Slant height: The distance from the apex of a cone or pyramid down its sloping surface to the base edge.
Net: A 3D solid unfolded flat — the easiest route to surface area.
Cross-section: The 2D shape you get by slicing a solid perpendicular to its length.
Apex: The single point at which a cone or pyramid comes together.

02Formulas

The formulas you need

Keep these on hand — every method below builds on them.

Prism (any cross-section)

V = A_{\text{base}} \cdot h

Covers cubes, cuboids and triangular prisms — find the cross-section area, then extrude.

Cylinder

V = \pi r^2 h, \quad SA = 2\pi r^2 + 2\pi rh

Two circular ends plus one rolled-out rectangle.

Cone

V = \tfrac{1}{3}\pi r^2 h, \quad SA = \pi r^2 + \pi r l

Volume uses the vertical height h; surface area uses the slant height l.

Sphere

V = \tfrac{4}{3}\pi r^3, \quad SA = 4\pi r^2

Radius only — a sphere has no height to supply.

Pyramid

V = \tfrac{1}{3}A_{\text{base}} h

Same one-third rule as the cone, with any polygon as the base.

Slant height of a cone

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

Recovered with the Pythagorean theorem when only r and h are given.

03Methods

How to solve surface area and volume

Pick the method that fits the problem in front of you.

When to use it: Any prism or cylinder — a solid with the same shape all the way through.

1Identify the cross-section: the face that repeats along the length of the solid.
2Find its area with the matching 2D formula, e.g. a circle of radius $3$ gives $9\pi$ .
3Multiply by the height or length: $V = 9\pi \cdot 10 = 90\pi$ .
4Report in cubic units — three lengths were multiplied together.

04Worked examples

Surface Area and Volume, solved step by step

From easy to hard — pick a problem to see its full solution.

\text{Cuboid: } 4 \times 5 \times 6

1Volume: multiply all three dimensions, $V = 4 \cdot 5 \cdot 6 = 120$ cubic units.
2Surface area: three pairs of faces, $2(4\cdot5) + 2(4\cdot6) + 2(5\cdot6)$ .
3 $SA = 40 + 48 + 60 = 148$ square units.

Try it on your own surface area and volume homework

Photo Math Solver reads a solid-geometry problem straight from a screenshot — a labelled cylinder, a cone with its slant height marked, a composite net — and shows which formula applies, where each measurement goes, and the answer with the right units. It keeps surface area and volume apart, which is where most of these questions are actually lost.

05Practice

Now you try

Work each one on paper first, then check your answer.

Problem 1

\text{Cube with side } 5

Problem 2

\text{Cylinder: } r = 2,\; h = 7

Problem 3

\text{Cone: } r = 6,\; h = 10

Problem 4

\text{Sphere with surface area } 100\pi

Problem 5

\text{Open-topped cylindrical tank: } r = 4,\; h = 9. \text{ Find } SA.

Stuck on your surface area and volume homework? Screenshot it

Solid-geometry questions hinge on which labelled measurement is which — radius or diameter, vertical height or slant. Screenshot the diagram with Photo Math Solver instead of retyping the numbers, and the solution names each measurement before it substitutes.

06Common mistakes

Where points get lost

Each of these shows up on real graded work — and each has a simple fix.

The mistake	Why it happens	The fix
Using the slant height in the cone volume formula instead of the vertical height.	Diagrams often label the slant because it is a visible edge, while the vertical height is an internal dashed line.	Volume always uses the perpendicular height $h$ ; only surface area uses the slant $l$ . If only $l$ and $r$ are given, recover $h = \sqrt{l^2 - r^2}$ first.
Dropping the $\frac{1}{3}$ for cones and pyramids.	The cylinder and prism formulas are used far more often and become the automatic response to "base × height".	Anything that tapers to a point holds one third of the solid that boxes it in. If your cone volume matches the cylinder around it, the factor is missing.
Answering with square units for a volume, or cubic units for a surface area.	The unit is attached at the end by habit rather than from what was actually multiplied.	Count the lengths in the calculation: three multiplied gives cubic units, faces added gives square units. Mark schemes routinely deduct for this alone.
Including both ends when the solid is open — a pipe, a tube, or an open tank.	The full formula $2\pi r^2 + 2\pi rh$ is memorised as one block and applied without reading the wording.	Read for "open", "tube", or "no lid" and build the surface area face by face instead of reciting the closed formula.
Taking a square root when reversing a sphere volume.	Almost every other reverse problem ends in a square root, so the cube gets treated the same way.	Match the root to the power: $r^3 = 27$ needs a cube root, giving $r = 3$ , not $\sqrt{27} \approx 5.2$ .

Photo Math Solver shows every intermediate step, so slips like these are easy to catch before they cost you marks.

07FAQ