Educerie · SAT · Math
Geometry and Trigonometry · GT.1 Area and volume
What you must be able to do
| You must be able to | What it looks like on the test |
|---|---|
| Use the given formulas correctly | Radius rather than diameter, height rather than slant |
| Work backwards from an area or volume | …the volume is 400; find the radius |
| Handle composite shapes | Add, or subtract a hole |
| Scale areas and volumes | By k² and k³, not by k |
| Convert units inside a formula | cm to m before cubing, not after |
1The idea in one paragraph
Every formula you need is printed on the reference sheet, so the test cannot be examining whether you memorised them. What it examines is whether you put the right number into the right slot, whether you can run the formula backwards, and whether you know that doubling a length multiplies volume by eight. The arithmetic is easy and the setup is where the points are.
Write down which letter is which before substituting. Half the wrong answers in this unit come from using the diameter as the radius.
2The formulas you are given, and their traps
| Shape | Formula | The trap |
|---|---|---|
| Circle | A = πr², C = 2πr | Diameter given instead of radius |
| Triangle | A = ½bh | Height must be perpendicular, not a slanted side |
| Cylinder | V = πr²h | Same radius trap |
| Sphere | V = (4/3)πr³ | Cubing before halving the diameter |
| Cone | V = (1/3)πr²h | Forgetting the third |
| Pyramid | V = (1/3)lwh | Forgetting the third |
3Running a formula backwards
A cylinder has volume 360π cm³ and height 10 cm. Find its radius.
Keep π symbolic as long as possible: it usually cancels, and the answer choices are often written in terms of π. Converting to a decimal early costs accuracy and time.
4Composite shapes
Break the shape into pieces you have formulas for, then add — or subtract, when something has been removed.
Shade the pieces mentally before calculating. Most errors here are counting a region twice or forgetting to subtract.
5Scaling: the k² and k³ rule
If every length is multiplied by k:
- Areas are multiplied by k²
- Volumes are multiplied by k³
A model at 1 : 4 scale has 1/16 the surface area and 1/64 the volume of the real thing. A question about a scaled-up tank or a shrunken model is nearly always testing this, and the wrong answers are the ones that used k.
6Units inside formulas
Convert lengths before substituting. A length in centimetres cubed gives cm³, and converting afterwards means dividing by 1,000,000 rather than by 100 — a mistake that produces an answer a million times too big and still looks like a number.
Where points are lost
- Diameter for radius. The single most common error in this unit.
- Using a slant length as a height.
- Forgetting the third in a cone or pyramid.
- Scaling area or volume by k instead of k² or k³.
- Converting units after cubing.
- Rounding π early when the options are written in terms of π.
Work it right
- Write the formula from the reference sheet.
- Label every number in the question with which letter it is.
- Convert units before substituting.
- Solve, keeping π symbolic.
- Check the units of the answer: cm² for an area, cm³ for a volume.
Try it
Q1. A circle has a diameter of 14 cm. What is its area, in cm²?
A) 14π B) 49π C) 98π D) 196π
Q2. A cylinder has a radius of 5 cm and a volume of 200π cm³. What is its height, in cm? (Type your answer.)
Q3. A cube's edges are each tripled. The volume of the new cube is how many times the volume of the original?
A) 3 B) 9 C) 18 D) 27
Q4. A rectangular garden 12 m by 9 m has a circular pond of radius 3 m cut out of it. What is the remaining area, in m²?
A) 108 − 3π B) 108 − 9π C) 108 − 6π D) 99π
Q5. A cone has radius 6 cm and height 10 cm. What is its volume, in cm³?
A) 60π B) 120π C) 180π D) 360π
In one breath
Every formula is printed for you, so the test is checking the setup instead: the radius rather than the diameter, the perpendicular height rather than a slant, and the third that a cone and a pyramid need. Keep π symbolic because it usually cancels, convert units before you substitute rather than after cubing, and remember that multiplying every length by k multiplies areas by k² and volumes by k³.
Answers
Q1. B — 49π. halve the diameter first
D uses the diameter as the radius. A and C confuse area with circumference.
Q2. 8. run the formula backwards
Q3. D — 27. volume scales by k³ 3³ = 27. A scales by k and B by k², which is the surface-area answer.
Q4. B — 108 − 9π. subtract the hole
A uses the radius instead of its square; C uses the circumference.
Q5. B — 120π. a third of πr²h
D omits the third; A and C slip on the radius.
Educerie · written from the published College Board* Assessment Framework for the Digital SAT Suite *(Math, Geometry and Trigonometry, skill/knowledge testing point "Area and volume"). All questions and explanations are original Educerie text. Last reviewed 12 September 2026.
Check your understanding
The main ideas of this note. Tick each one you could do now, in an exam, without looking back up. Anything you cannot tick yet is the part to read again.