Area & volume
These questions hand you a shape or solid and ask for a measurement — area, perimeter, volume, or surface area. The SAT gives you a formula reference sheet, so the work is recognizing which formula fits and setting it up correctly.
What College Board tests
Area and perimeter of polygons, area and circumference of circles, area from coordinates, volume and surface area of solids (cylinders, cones, spheres, prisms), and composite or shaded regions where you combine or subtract shapes. Many come as real-world word problems.
Formulas worth knowing cold
The reference sheet has these, so don't memorize blindly — practice matching the formula to the shape and reading dimensions carefully. Most mistakes come from using a radius where a diameter was given, or a slant height where a vertical height was needed.
Six worked examples covering the full range of area & volume types. Read the approach, try it, then tap Show the full solution.
1 · Area of a triangle from coordinates
A triangle has vertices at , , and in the xy-plane, as shown. What is the area, in square units, of the triangle?
Approach Two sides lie along the axes, so they're perpendicular — the legs are the base and height. Read their lengths straight off the coordinates, then use .
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Answer: B
Why the other choices are wrong
A Uses base 5 instead of 6.
C Forgets the one-half (that's ).
D Doubles the base-times-height product.
2 · Area & circumference of a circle
A circle has a radius of 5. What are its area and circumference, respectively?
Approach Area uses ; circumference uses . Keep them straight — area squares the radius, circumference doesn't.
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Answer: B
Why the other choices are wrong
A Swaps the two values.
C Uses for area.
D Uses for circumference.
3 · Volume of a cylinder
A right circular cylinder has a radius of 3 and a height of 10. What is its volume?
Approach Cylinder volume is the circular base area times the height: . Square the radius first, then multiply by the height.
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Answer: C
Why the other choices are wrong
A Uses instead of ().
B Uses the diameter or doubles incorrectly.
D Squares the whole product (uses ).
4 · Volume of a cone
A cone has a radius of 6 and a height of 8. What is its volume?
Approach A cone is one-third of the cylinder with the same base and height: . The "8" is the vertical height, which is what the formula needs.
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Answer: B
Note the "6" and "8" here are the radius and the vertical height — a right angle at the cone's axis. If a problem gives the slant height instead, use the Pythagorean theorem to recover the vertical height first.
Why the other choices are wrong
A Uses without squaring ().
C Uses half instead of one-third.
D Forgets the one-third (computes the full cylinder).
5 · A composite region
A rectangular sheet measures 12 by 8. A square notch measuring 4 by 4 is cut from one corner, as shown. What is the area of the remaining region?
Approach Find the full rectangle's area, then subtract the square that was removed. Composite-region problems are almost always "big shape minus small shape."
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Answer: A
Why the other choices are wrong
B Subtracts only a piece.
C Subtracts the notch's perimeter instead of its area.
D Forgets to subtract the notch at all.
6 · A volume word problem
A cylindrical tank with radius 2 feet and height 5 feet is filled with water. If the water is poured into cubical containers each holding cubic feet, how many containers can be completely filled? (Use .)
Approach Find the tank's volume, then divide by the volume each container holds. The cancels, leaving a clean count.
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Answer: B
Why the other choices are wrong
A Uses instead of .
C Doubles the tank volume.
D Forgets to divide by the container size.