One for All SAT › SAT notes › Geometry and Trigonometry › Area and volume
Area and volume: SAT Math notes
Find the area, perimeter, volume or surface area of a shape, often by working backward from one measurement to another, and see how these change when a shape is scaled up or down.
Study it with worked examples and 8 practice questionsPractise Area and volume questions
What this topic tests
These questions are about how much space a shape covers or fills. Area counts the square units inside a flat shape. Perimeter (called circumference for a circle) is the distance around it. Volume counts the cubic units inside a solid, and surface area adds up the areas of all its faces.
You don't have to memorise most of the formulas. The SAT reference sheet gives the area of a circle, rectangle and triangle, the circumference of a circle, and the volume of a rectangular prism, cylinder, sphere, cone and pyramid. Your job is to pick the right formula, put each number in the right place, and solve. Many questions run a formula backward: they give you the volume and ask for the radius, or give you the area and ask for a side.
The other big idea is scaling. When every length of a shape is multiplied by the same number, the area and the volume grow much faster than the lengths do. Once you know the rule, these questions take seconds.
Key ideas
- Rectangle: A=lw. Triangle: A=1/2bh. Circle: A=πr2 and C=2πr. All of these are on the reference sheet.
- Prism or cylinder: volume = base area × height, so V=lwh or V=πr2h. Cone or pyramid: one third of that, V=1/3πr2h or V=1/3lwh. Sphere: V=4/3πr3.
- If a question gives a diameter, halve it before you use any formula with r in it.
- Surface area is not on the reference sheet. Add up the area of every face. A cube with edge s has six square faces, so its surface area is 6s2.
- If every length is multiplied by k, perimeters are multiplied by k, areas by k2, and volumes by k3.
- Area is measured in square units and volume in cubic units. The units in the question tell you which one you're finding.
The 3 kinds of Area and volume question
Sorted from every Area and volume question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Area and perimeter of flat shapes
How to spot it. The question is about a flat shape: a rectangle, square, triangle or circle, or two of them together. It asks for an area, a perimeter or circumference, or a missing side when the area is given. Sometimes the shape is drawn on a coordinate grid.
How to answer it:
- Write the formula for the shape: lw for a rectangle, s2 for a square, 1/2bh for a triangle, πr2 or 2πr for a circle.
- Find every length the formula needs. Read the wording slowly, because one length is often described using another, like a length that is 40 more than the width.
- If the shape sits on a grid, get its side lengths from the coordinates. Count across and up, and use the Pythagorean theorem for slanted sides.
- Substitute and solve. If the area is given and a side is missing, set the formula equal to the area and solve for the side.
- Check that you answered what was asked: area or perimeter, the side or the side squared.
The trap. Mixing up area and perimeter. For a rectangle, 2(l+w) is the perimeter and lw is the area, and the wrong one is almost always a choice. Also watch the 1/2 in the triangle formula: forgetting it gives exactly double the right answer.
2. Volume and surface area of solids
How to spot it. The question names a 3D shape: a box (rectangular prism), cube, cylinder, cone, pyramid, sphere or hemisphere. It asks for a volume or surface area, or gives you one of those and asks for a radius, height or edge.
How to answer it:
- Name the solid and write its volume formula from the reference sheet. For surface area, list the faces and add up their areas.
- Collect the measurements. Turn a diameter into a radius, and a circumference into a radius using C=2πr.
- Substitute what you know. If the volume is given, set the formula equal to it.
- Solve for the unknown. Divide out π and any other factors first, then take a square root or cube root at the end if you need one.
The trap. Stopping one step early. When you solve πr2h=V, you get r2 first, and that number is usually one of the wrong choices. Also remember the extra 1/3 for cones and pyramids, and halve any diameter.
3. Scale factors with similar figures
How to spot it. Two shapes are similar, or a shape is enlarged, shrunk or copied at a new size. Look for words like similar, times the length, twice the radius, or increased by 20%. You're asked how a perimeter, area or volume changes, or to work back from an area or volume ratio to a length.
How to answer it:
- Find the length scale factor k: a new length divided by the matching old length.
- Match the measurement to its power. Lengths and perimeters change by k. Areas, including surface areas, change by k2. Volumes change by k3.
- If you're given an area ratio, take its square root to get k. If you're given a volume ratio, take its cube root.
- Multiply or divide, checking which shape is bigger so your answer comes out the right size.
- If only some dimensions change (say the radius doubles but the height halves), put each change into the formula separately and multiply the effects together.
The trap. Using k for everything. If the sides double, the area doesn't double; it becomes 4 times as big. Going the other way, if one area is 9 times another, the sides are only 3 times as long.
Practise Area and volume
On One for All SAT every question has a step-by-step solution, worked both by hand and in Desmos, the calculator built into the digital SAT, and the notes have worked examples of each kind of question, then practice questions you can check as you go. It's free, and there's nothing to sign up for.
Open the Area and volume notes