SAT Math · Geometry and Trigonometry · Skill 1 of 4

Area and volume

Find the area of flat shapes and the volume and surface area of solids, work backwards to a missing length, and see exactly what happens when a shape is scaled up or down.

9 chapters plus a challengeAbout 40 minutesFree, no sign-up
Start with the scale lab →Jump to the challenge
Hi, I’m Io. The formulas are on the SAT reference sheet. The points come from knowing which one to use, and what scaling does to it.
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Grow a box, watch the numbers

Explore · 5 min

This box starts 3 by 2 by 2. The slider multiplies every length by the same scale factor k. Watch how the surface area and the volume respond. They grow much faster than the edges do.

Area formulas

Learn · 5 min

Area measures the flat space inside a shape, in square units. The SAT gives these on its reference sheet, but knowing them by heart saves time. In every one, the height meets the base at a right angle.

RectangleA = lw
TriangleA = ½bh
ParallelogramA = bh
TrapezoidA = ½(a + b)h
CircleA = πr² · C = 2πr

Answers with πThe SAT often leaves π in the answer, like 9π. In these lessons, when a box is followed by π, type only the number in front of it.

Composite shapes

Practise · 5 min

Most SAT area questions use a shape made from simpler ones. Two moves cover almost all of them: add the pieces, or take a big shape and subtract the part that is missing.

Worked example: an L-shaped room, 12 m by 9 m with a 4 m by 3 m corner cut out
  1. Treat it as the full rectangle: 12 × 9 = 108 m²
  2. The missing corner: 4 × 3 = 12 m²
  3. Subtract: 108 − 12 = 96 m²

Volume of solids

Learn · 6 min

Volume measures the space inside a solid, in cubic units. One idea covers most of it: volume = area of the base × height. A cone or pyramid holds exactly one third of the matching cylinder or prism.

Rectangular prismV = lwh
CylinderV = πr²h
ConeV = ⅓πr²h
PyramidV = ⅓lwh
SphereV = ⁴⁄₃πr³

Surface area

Learn · 5 min

Surface area is the total area of every outside face, in square units. Picture unfolding the solid flat and adding up the pieces.

  • Box: three pairs of matching faces, so SA = 2(lw + lh + wh). A cube with edge s has SA = 6s²
  • Cylinder: two circles plus a rectangle that wraps around. SA = 2πr² + 2πrh. The rectangle’s width is the circumference, 2πr

Working backwards

Practise · 5 min

Many SAT questions give the area or volume and ask for a length. Write the formula, put in everything you know, then solve for what is left.

Worked example: a cylinder has volume 72π cm³ and radius 3 cm. Find its height.
  1. Formula: V = πr²h
  2. Put in what you know: 72π = π × 3² × h = 9πh
  3. Divide both sides by 9π: h = 8 cm

The scaling rules

Sort · 5 min

When every length of a shape is multiplied by k, the new shape is similar to the old one. Then:

lengths × k  ·  areas × k²  ·  volumes × k³

A cube with edges twice as long needs 4 times the paint and holds 8 times the water. Sort each measurement for a solid whose lengths are all multiplied by k.

Area and volume in real life

Scenarios · 6 min

Each tab is a situation you could meet on test day. Answer every question to finish the chapter.

Spot the trap

Detective · 3 min

Each solution has one faulty step. All three slips appear as wrong answer choices on the SAT.

Final challenge

8 questions · 10 min

Key ideas to carry in
HeightAlways at a right angle to the base.
Composite shapesAdd the pieces, or subtract the missing part.
VolumeBase area × height. Cones and pyramids: one third of that.
Surface areaAdd the area of every face.
Work backwardsPut in what you know, then solve.
ScalingLengths × k, areas × k², volumes × k³.
Challenge