Compute the volume and surface area of prisms, cylinders, cones, spheres, and pyramids using the reference sheet formulas, and solve real-world problems involving capacity and material.

What this topic is asking

The Geometry category requires you to compute volume and surface area of solids (the G-GMD standards). On the Grade 10 MCAS you handle prisms, cylinders, cones, spheres, and pyramids, applying the reference sheet formulas to capacity and material problems. All the volume formulas are on the reference sheet, so the credit is for choosing the right formula, substituting correctly, and reporting with appropriate units.

Volume formulas

Volume measures the space inside a solid, in cubic units. The reference sheet provides:

Here $B$ is the area of the base and $h$ the height. For a prism, find the base area first (a rectangle, triangle, or other polygon), then multiply by the height. The crucial pattern: a cone is one-third of a cylinder with the same base and height, and a pyramid is one-third of the matching prism, which is why those formulas carry the $\frac{1}{3}$. Forgetting that factor is the single most common error.

Surface area

Surface area is the total area of all the faces of a solid, in square units. For a solid with flat faces, add the areas of every face; the reference sheet provides the total surface area formulas alongside the volumes.

• A rectangular prism has surface area $2(lw + lh + wh)$: the six faces in three matching pairs.
• A cylinder has surface area $2\pi r^2 + 2\pi r h$: the two circular ends plus the rectangle that wraps around (its width is the circumference).
• A sphere has surface area $4\pi r^2$.

The distinction the MCAS tests is dimension: surface area is square units (covering material, paint, wrapping), while volume is cubic units (capacity, filling). Reading whether the problem asks for the amount to fill (volume) or to cover (surface area) decides which to compute.

Real-world problems and units

Capacity and material problems are volume and surface-area problems in context. "How much water fills the tank" is a volume in cubic units or a capacity (liters, gallons). "How much paint covers the box" is a surface area in square units. Always attach the correct unit, and when a decimal is required, use $\pi \approx 3.14$ from the reference sheet, rounding at the end.

For composite solids (a cylinder topped by a cone, say), find each part's volume separately and add them. Subtraction is used when a hole is removed.

Scaling solids by a factor

When a solid is scaled by a linear factor $k$, its volume scales by $k^3$, because volume is a product of three lengths. So doubling every dimension of a box ($k = 2$) makes it hold $2^3 = 8$ times as much, not twice as much. This often appears as a "how many times bigger" question, and the answer is the cube of the linear factor. By the same logic, surface area scales by $k^2$. A common error is to scale volume by the linear factor or its square instead of its cube.

Working backward from a given volume

Sometimes the MCAS gives a volume and asks for a missing dimension. Substitute into the formula and solve. If a cylinder has volume $100\pi$ and radius 5, then $\pi(5)^2 h = 100\pi$, so $25h = 100$ and $h = 4$. The same rearrangement works for any solid: write the formula, plug in the known volume and dimensions, and solve for the unknown, treating it like any equation.

Try this

Q1. A sphere has radius 3. Find its volume in terms of $\pi$.

• Cue. $\frac{4}{3}\pi(3)^3 = \frac{4}{3}\pi(27) = 36\pi$.

Q2. A rectangular prism is 2 by 3 by 5. Find its volume.

• Cue. $V = 2 \times 3 \times 5 = 30$ cubic units.