Circumference, Area, and Volume
Lessons in this chapter
About Circumference, Area, and Volume
Circumference, Area, and Volume applies every geometric shape covered so far to real measurement: how far around, how much surface, and how much space.
Circumference and Arc Length and Areas of Circles and Sectors apply the circle formulas from Circles to real distance and area problems, including partial arcs and sectors, while Areas of Polygons extends area calculations to any polygon, not just circles and quadrilaterals.
Three-Dimensional Figures introduces solid shapes and their nets, before Volumes of Prisms and Cylinders and Volumes of Pyramids cover the volume formulas for straight-sided and pointed solids, including the one-third relationship between a pyramid and the matching prism.
Surface Areas and Volumes of Cones and Surface Areas and Volumes of Spheres close the chapter with the two remaining curved solids, applying circle-area formulas to their curved surfaces.
Each of the eight lessons here opens as a Geometry worksheet in the page, including Areas of Circles and Sectors, Volumes of Prisms and Cylinders and Surface Areas and Volumes of Spheres. Questions are scored as you answer them, which is the practical way to get the geometry formulas straight, since most errors in this chapter are a radius used where a diameter was given rather than anything conceptual. All eight are free to try, with no login needed for your first 3.
Frequently asked questions
Circumference and Arc Length and Areas of Circles and Sectors treat a part of a circle as a fraction of the whole, so one idea covers both arc length and sector area. Areas of Polygons extends area to regular polygons, and Three-Dimensional Figures introduces solids and their nets before Volumes of Prisms and Cylinders, Volumes of Pyramids and Surface Areas and Volumes of Cones and Surface Areas and Volumes of Spheres work through each solid in turn. The chapter sits in the high school Geometry conceptual category (G).
Circumference is C = 2πr (or πd), and area is A = πr²: both use only the radius (or diameter).
Find what fraction of the full circle the sector’s central angle represents (angle divided by 360°), then multiply that fraction by the full circle’s area.
Break the polygon into simpler shapes (triangles, rectangles) with known area formulas, calculate each piece’s area, and add them together.
A pyramid's volume is always exactly one-third of a prism's volume, given the same base area and height.
The same one-third relationship holds: a cone's volume is exactly one-third of a cylinder's volume, given the same base radius and height.