On the test
Geometry and Trigonometry is about 15% of the Math section, around 2 to 4 questions in each 22-question module, spread through the module from easier to harder.
5 to 7 of the 44 questions (about 2 to 4 in each module). Spread through the module, easier first and harder at the end.
How I approached it
A small story from Aime
When I look at an area or volume question, I first ask one thing: what am I really counting? Think about tiling a classroom floor. You count how many square tiles cover the floor. That is area. Now think about filling a water tank at home. You count how many small cubes of water fit inside. That is volume. You do not count tile by tile. You measure the length and width, and multiply. The SAT works the same way. It gives you a shape and some measurements, and asks how much space it covers or holds. You do not need to memorise a mountain of rules. You need to know what is being counted, find the formula on the reference sheet, and carefully put your numbers in.
Area and volume questions are friendly because the formulas are given to you in Bluebook. Your job is only to choose the right one and not rush the arithmetic. You can learn that.
What is inside this part
Finding how much surface a rectangle, triangle, circle or similar shape covers.
Rectangles and squares · Triangles (½bh) · Circles (πr²) · Shaded regions and combined shapes
Finding how much space is inside a 3D shape such as a prism, cylinder, cone, sphere or pyramid.
Rectangular prisms and cubes · Cylinders · Cones and spheres · Pyramids
Using a given area or volume to find a length, radius or height.
Solving for a height or radius · Pouring liquid from one container to another · Surface area of a cube or prism
Working out what happens to area or volume when lengths are multiplied.
Doubling or tripling a radius · Comparing two solids with related sizes
How to know a question is from this part
A shape is described with lengths and the question asks for an area, a volume, or a missing length.
- Words like area, volume, surface area, cylinder, cone, sphere, prism, tank, box
- A question asking 'in terms of π' or giving answers with π in them
- 'Water is poured' or 'filled' from one container into another
- A shaded region inside a bigger shape
- 'What happens to the volume if the radius is doubled?'
The lesson
1. What area and volume mean
Area is the amount of flat surface a shape covers. We measure it in square units, like square metres (m²). If a floor is 5 metres long and 4 metres wide, you can place 20 squares of 1 metre by 1 metre on it, so the area is 20 m².
Volume is the amount of space inside a solid shape (a 3D shape). We measure it in cubic units, like cubic centimetres (cm³). A box that is 5 cm long, 4 cm wide and 3 cm tall holds 5 × 4 × 3 = 60 little cubes, so its volume is 60 cm³.
Perimeter and surface area are different. Perimeter is the distance around a flat shape (like a fence). Surface area is the total area of all the outside faces of a solid (like the paper needed to wrap a box). Always check which word the question uses.
2. The formulas you need
Bluebook gives you a reference sheet with these formulas, so you do not have to memorise them. But you must know how to read them, and it helps to know them by heart so you work faster.
Flat shapes: rectangle A = lw. Triangle A = ½bh (b is the base, h is the height, and the height makes a right angle with the base). Circle A = πr², and circumference C = 2πr. Trapezoid is not on the sheet, but A = ½(b₁ + b₂)h, the average of the two parallel sides times the height.
Solids: rectangular prism V = lwh. Cylinder V = πr²h. Sphere V = (4/3)πr³. Cone V = (1/3)πr²h. Pyramid V = (1/3)lwh. Notice the pattern: a cone is one third of the cylinder with the same base and height, and a pyramid is one third of the prism.
3. What the question is really asking
Almost every area and volume question is a small story: one number is missing, the others are given. Your work is to match the story to a formula, then put in the numbers.
Read the last sentence first. It tells you what to find (area, volume, a side, a radius, a height). Then read the rest and write down every length you are given, with its name (r = 6, h = 10).
Watch the words: diameter is twice the radius. Height of a triangle is not the slanted side. The question may ask for your answer in terms of π, which means you leave π as a symbol, like 36π, and do not turn it into a decimal.
4. The method, step by step
Step 1: Say what is being asked in your own words. Step 2: Write the formula. Step 3: Replace each letter with its number, one by one. Step 4: Calculate carefully, squaring before multiplying. Step 5: Check the units and ask: does this size make sense?
For a radius, you square only the radius, not the π. In πr² with r = 6, you do 6² = 36 first and then write 36π.
A cylinder has radius 3 and height 10. What is its volume?
Show the steps
- The formula is V = πr²h.
- Put in the numbers: V = π × 3² × 10.
- 3² = 9, so V = π × 9 × 10 = 90π.
90π
5. Working backwards
Sometimes the volume or area is given and you must find a length. Write the formula, put in what you know, and solve for the missing letter, like a normal equation.
Another common idea: water is poured from one container into another. The amount of water (volume) stays the same. So you find the volume in the first container, then use it in the second container's formula.
The area of a triangle is 54 and its base is 12. What is its height?
Show the steps
- A = ½bh, so 54 = ½ × 12 × h.
- 54 = 6h.
- h = 9.
9
6. When a length is scaled
If you double the radius of a circle, the area does not double. It becomes 4 times bigger, because r is squared. If you double every length of a solid, the volume becomes 8 times bigger. Squares and cubes grow fast.
So for a change question, compare the new formula to the old one. Put the new numbers in, cancel what stays the same, and see what multiplier is left.
Cylinder A has radius r and height h, and its volume is 40. Cylinder B has radius 2r and height h. What is the volume of B?
Show the steps
- Volume A = πr²h = 40.
- Volume B = π(2r)²h = π × 4r² × h = 4πr²h.
- So B = 4 × 40 = 160.
160
7. Common traps
Trap 1: using the diameter where the formula needs the radius. Halve it first. Trap 2: forgetting to square the radius, or squaring the π as well. Trap 3: using the slanted side of a triangle or cone as the height. The height must be straight up, at a right angle to the base.
Trap 4: forgetting the one third for cones and pyramids. Trap 5: mixing units, such as metres and centimetres. Convert first. Trap 6: finding the area when the question asks for the perimeter, or the other way round.
8. Combined shapes
Some questions show a shape made of two shapes, or a shape with a hole. For a shaded region, find the area of the big shape and subtract the area of the part that is cut out. For a combined shape, add the parts.
Example: a square with side 10 has a circle of radius 5 cut out of it. The shaded area is 100 − 25π. Keep it in this form if the options are in terms of π.
See it
| Shape | What we find | Formula |
|---|---|---|
| Rectangle | Area | lw |
| Triangle | Area | ½ × b × h |
| Circle | Area / circumference | πr² / 2πr |
| Rectangular prism | Volume | lwh |
| Cylinder | Volume | πr²h |
| Cone | Volume | ⅓ × πr²h |
| Sphere | Volume | (4/3)πr³ |
| Pyramid | Volume | ⅓ × lwh |
- 1Read the last sentence: what must I find?
- 2List every given length with its name
- 3Write the formula
- 4Put in numbers, square the radius first
- 5Check units and common sense
The base is a circle with area πr². Stack that circle h units high and you get the volume.
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