GeometryGeometryHigh school

Area and Volume

Areas of polygons and volumes of prisms, cylinders, cones and spheres.

Area measures the space a flat shape covers, in square units; volume measures the space a solid occupies, in cubic units. Many formulas come from one idea: a prism or cylinder has volume equal to base area times height, and a pyramid or cone with the same base and height holds exactly one third of that.

Surface area is the total area of all faces, which you can find by unfolding the solid into a net. When a figure is scaled by a factor $k$ , lengths scale by $k$ , areas by $k^2$ and volumes by $k^3$ , which explains why doubling a box's dimensions multiplies its capacity by eight.

Key terms

Volume of a sphere: Four-thirds pi r cubed.
$V = \tfrac{4}{3}\pi r^3$
Volume of a cone: One third of base area times height.
$V = \tfrac{1}{3}\pi r^2 h$

Key ideas

Rectangle $lw$ , triangle $\frac{1}{2}bh$ , trapezoid $\frac{1}{2}(b_1 + b_2)h$ , circle $\pi r^2$ .
Cylinder $V = \pi r^2 h$ ; cone $V = \frac{1}{3}\pi r^2 h$ ; sphere $V = \frac{4}{3}\pi r^3$ .
Composite shapes: split into simple parts, then add or subtract.
Keep units consistent before multiplying.

Worked example

A cylinder has radius 3 cm and height 10 cm. Find its volume and total surface area.

Volume: $\pi (3)^2 (10) = 90\pi \approx 282.7$ cm³.
Two circular ends: $2\pi (3)^2 = 18\pi$ .
Curved side: $2\pi (3)(10) = 60\pi$ .
Surface area: $78\pi \approx 245.0$ cm².

Answer: $V = 90\pi$ cm³, $SA = 78\pi$ cm²

Common mistakes

Mixing radius and diameter.
Forgetting the $\frac{1}{3}$ for cones and pyramids.
Reporting volume in square units.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Find the volume of a box 4 by 5 by 6.
Show answer
$120$
2.Find the volume of a sphere of radius 3, to one decimal.
Show answer
$113.1$
3.Find the area of a trapezoid with bases 7 and 11 and height 4.
Show answer
$36$

Key terms

Key ideas

Worked example

Common mistakes

Practice questions

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