Compute the perimeter and area of triangles, rectangles, parallelograms, trapezoids and circles, and the surface area and volume of prisms, cylinders and other common solids (Geometry, Integrating Essential Skills).

What this topic is asking

This topic tests the measurement formulas for common 2-D shapes (area, perimeter) and 3-D solids (surface area, volume). It is high-frequency on the ACT and feeds many Integrating Essential Skills word problems. The skill is recalling the right formula and, for composite figures, breaking the shape into pieces.

Area and perimeter of 2-D shapes

The core area formulas to memorise.

The height in a triangle or parallelogram is the perpendicular distance to the base, not a slanted side.

Volume of solids

Volume measures the space inside a solid.

The general rule for any prism or cylinder is $V = (\text{base area}) \times \text{height}$, which specialises to $lwh$ for a box and $\pi r^{2} h$ for a cylinder.

Surface area

Surface area is the total area of a solid's outer faces. For a rectangular box with dimensions $l$, $w$, $h$, it is $2(lw + lh + wh)$, the sum of the areas of the three pairs of faces. For a cylinder, it is the curved side ($2\pi r h$) plus the two circular ends ($2\pi r^{2}$). Surface area is an area (square units); volume is a volume (cubic units), and confusing the two is a frequent ACT trap. Reading whether the question asks how much material covers the solid (surface area) or how much fits inside (volume) tells you which to compute.

Composite figures

Many ACT questions show a figure made of two or more simple shapes, such as a rectangle with a semicircle on one end, or an L-shaped region. Break it into known pieces, compute each separately, and add (for combined regions) or subtract (for a hole or cut-out). For a shaded region, it is often the area of a large shape minus a smaller one inside it. Labelling each piece and tracking which you add or subtract keeps a composite problem organised.

Cones, spheres and pyramids

Beyond prisms and cylinders, the ACT may use a cone, sphere or pyramid, whose volume formulas it usually provides if needed but which are worth recognising. A cone has volume $\frac{1}{3}\pi r^{2} h$ (one third of the cylinder with the same base and height). A sphere has volume $\frac{4}{3}\pi r^{3}$ and surface area $4\pi r^{2}$. A pyramid has volume $\frac{1}{3}(\text{base area})\times \text{height}$, the same one-third relationship as the cone. The recurring idea is that a pointed solid (cone, pyramid) holds one third of the matching prism or cylinder.

Effect of changing dimensions

A subtle ACT question asks how a measure changes when a dimension changes. Because area depends on two lengths, doubling one side doubles a rectangle's area, but doubling all sides multiplies it by four. Volume depends on three lengths, so doubling every edge of a box multiplies the volume by eight. Reading whether one dimension or all of them change, and counting how many length factors the measure depends on, prevents the common error of scaling area or volume by the wrong amount.

Why formula recall and decomposition win

Most points here come down to two things: recalling the correct formula (especially the $\frac{1}{2}$ on a triangle and the average of the bases on a trapezoid) and, for harder figures, decomposing into simple shapes. Tracking units also matters: area is square units, volume is cubic units, so an answer in the wrong dimension signals a formula slip. A quick check that the formula matches the question (area versus volume, surface versus inside) prevents the most common mistakes.

Try this

Q1. A trapezoid has parallel sides 6 and 10 and height 4. Find its area. [1 point]

• Cue. $\frac{1}{2}(6 + 10)(4) = \frac{1}{2}(16)(4) = 32$.

Q2. A cube has edge length 5. Find its volume. [1 point]

• Cue. $V = s^{3} = 5^{3} = 125$.