8.1 Area & Volume Formulas

Area and Volume From the Reference Sheet

Every Geometry Regents booklet includes the Next Generation Geometry Reference Sheet, and this domain rewards students who use it fluently. The sheet lists area and volume formulas, but it deliberately omits surface-area formulas for curved solids, so you must know how to build those yourself. Getting comfortable with B, the area of the base, is the single most useful habit here, because prisms, cylinders, cones, and pyramids all reference it.

Area Formulas You Reach For First

For polygons the base and height must be perpendicular; a slanted side is never the height. For circles, keep π exact (leave answers such as 36π) unless a question tells you to round.

Worked area example. A trapezoidal window has parallel sides of 8 in and 12 in and a perpendicular height of 5 in. Its area is A = ½(8 + 12)(5) = ½(20)(5) = 50 in². Averaging the two bases first, then multiplying by the height, is faster than splitting the trapezoid into a rectangle and triangle. Area answers always carry square units.

Volume Formulas — the Bh Family

The reference sheet groups volumes so you can see the pattern. A prism and a cylinder both use V = Bh. A pyramid and a cone both use V = ⅓Bh — exactly one-third of the prism or cylinder that shares the same base and height. A sphere stands alone.

Worked Volume Examples

Cylinder. Radius 3, height 5. The base is a circle, so B = π(3²) = 9π. Then V = Bh = 9π · 5 = 45π cubic units. A common trap is to write π(3)(5) and forget to square the radius.

Cone. A cone with base area 36π and height 10 gives V = ⅓Bh = ⅓(36π)(10) = 120π. Because the base area is already supplied, you do not re-derive it — just take one-third of Bh.

Prism. Base area 18 cm² and height 7 cm: V = Bh = 18 · 7 = 126 cm³. Volume units are always cubic.

Sphere. Radius 3: V = (4/3)π(3³) = (4/3)π(27) = 36π.

Surface Area — Build It, Don't Look It Up

Surface-area formulas for cones, pyramids, and spheres are not on the reference sheet, so memorize the essentials or reconstruct them.

• Prism / cylinder: total surface area = lateral area + 2·(base). A cylinder's lateral area is 2πrh (the label unrolls into a rectangle of width 2πr and height h), so its total is 2πrh + 2πr².
• Cone: lateral area = πrℓ, where ℓ is the slant height (not the vertical height). Total = πrℓ + πr².
• Sphere: surface area = 4πr² — exactly four great-circle areas.

Watch the difference between vertical height h (used for volume) and slant height ℓ (used for cone surface area). If a problem gives radius and height, find the slant with the Pythagorean theorem: ℓ = √(r² + h²).

Worked surface-area example. A closed cylinder has radius 3 and height 5. Its lateral area (the curved side) is 2πrh = 2π(3)(5) = 30π. Adding the two circular ends, each πr² = 9π, gives a total surface area of 30π + 2(9π) = 48π square units. Notice the same cylinder had volume 45π (cubic) — volume and surface area are different measurements with different units, so never interchange them.

Composite Solids

Many measurement questions stack two solids. Add the volumes when solids sit on top of each other and subtract when one is carved out. Example: a prism with base area 20 and height 6 topped by a pyramid on the same base with height 9. The prism gives Bh = 20·6 = 120; the pyramid gives ⅓Bh = ⅓(20)(9) = 60. Total volume = 120 + 60 = 180 cubic units. Keep base areas and heights labeled so you do not accidentally apply the ⅓ factor to the prism.

A carved example: a cube 10 cm on each edge with a cylindrical hole of radius 2 cm drilled all the way through has volume 10³ − π(2²)(10) = 1000 − 40π cm³. Leaving 40π in exact form is expected unless the problem says to approximate. Recognizing whether the second solid is added or removed is often where credit is won or lost on Part II–IV items.