A cylinder is a three-dimensional solid with two parallel, equal circular bases joined by one curved surface, like a soup can. Its size is set by the radiusr of a base and the height h between the bases. Its volume is π r2h, the base area times the height.
A cylinder is a circle stacked up to a height h. That is why its volume is the circle’s area, π r2, times h.
Formal definition and notation
A right circular cylinder of radius r and heighth is the solid swept out when a disk of radius r moves straight along a segment of lengthh perpendicular to it. Its bases are congruentcircles in parallel planes, and the segment joining their centers (the axis) is perpendicular to both. If the axis is slanted, the cylinder is oblique; its volume is still π r2h, with h the perpendicular height.
A circle is flat (2-D). A cylinder is a solid (3-D) whose two ends are circles.
Cylindervs.Prism
A prism has polygon bases, such as rectangles or triangles, and flat sides. A cylinder has circular bases and one curved side. Both use V = base area × h.
Quick check: 3 questions
Question 1 of 3
What is the volume of a cylinder with radius 5 m and height 2 m?
Show answer and explanation
Answer: D.π(5)2(2) = 50π. 20π is the lateral area 2π rh, 10π forgets to square the radius, and 100π uses the diameter as the radius with height 1.
Question 2 of 3
The label on a can covers only its curved side. The can has radius 4 cm and height 11 cm. What is the label’s area?
Show answer and explanation
Answer: A. Lateral area = 2π rh = 2π(4)(11) = 88π. 176π is the volume, 120π adds both bases (the total surface area), and 44π forgets the 2.
Question 3 of 3
How many curved surfaces, flat faces and vertices does a cylinder have?
Show answer and explanation
Answer: A. A cylinder has one curved side, two flat circular bases, and no corners. One flat face and one vertex describe a cone.
Cylinder: frequently asked questions
How do you find the volume of a cylinder?
Multiply the area of the circular base by the height: V = π r2h. Square the radius, multiply by π, then multiply by the height. For a radius of 2 ft and a height of 9 ft: π × 4 × 9 = 36π ≈ 113.1 cubic feet. If you are given the diameter, halve it first.
Is a cylinder a prism?
Not by the strict definition, because a prism’s bases must be polygons and its sides flat. A cylinder’s bases are circles and its side is curved. But the two shapes behave the same way for volume: both equal base area times height. That is why many textbooks teach them together.
What is the lateral area of a cylinder?
The lateral area is the area of the curved side only, without the two circular bases: L = 2π rh. Think of peeling off a can’s label and laying it flat. It is a rectangle with length equal to the circumference 2π r and width equal to the height h. Total surface area adds the two bases, 2π r2.
RadiusThe radius is a line segment from the center of a circle to any point on the circle, or the length of that segment. The radius is half the diameter: if the diameter is 10, the radius is 5. All radii in the same circle have equal length, and r is the radius in C = 2π r.Geometry · Grade 6, Grade 7, Grade 8VolumeVolume measures the amount of space inside a three-dimensional figure, in cubic units such as cubic centimeters or cubic feet. You can find it by counting the unit cubes that fill the solid. For a rectangular prism, V = l × w × h; for any prism, it equals the area of the base multiplied by the height.Geometry · Grade 5, Grade 6, Grade 7HeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3CongruentCongruent means exactly the same size and shape. Two segments are congruent when they have the same length, two angles when they have the same measure, and two figures when one can be turned, flipped or slid onto the other. The symbol is ≅: AB ≅ CD reads “segment AB is congruent to segment CD.”Geometry · Grade 8, GeometryCircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r2.Geometry · Kindergarten, Grade 1, Grade 7Surface AreaSurface area is the total area of all outside faces of a three-dimensional figure, measured in square units. To find it, unfold the solid into a flat net, find the area of each face and add them. A box with length l, width w and height h has surface area 2lw + 2lh + 2wh.Geometry · Grade 6, Grade 7, Grade 8SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2CircumferenceCircumference is the distance around a circle, found by multiplying the diameter by pi, or the radius by 2 pi: C = π d = 2π r. It is the circle’s perimeter, measured in units of length. A circle with diameter 10 has a circumference of about 31.4, because π ≈ 3.14.Geometry · Grade 6, Grade 7, Grade 8ConeA cone is a three-dimensional solid with one flat circular base and a curved surface that narrows to a single point, the apex, like a party hat. Its volume is one third of a cylinder with the same base and height: V = 13π r2h, where r is the base radius and h the height.Geometry · Kindergarten, Grade 1, Grade 8Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5DiameterThe diameter of a circle is the distance straight across it through the center. It is twice the radius, d = 2r, and it is the longest chord any circle has. The word also names the line segment itself: a segment that passes through the center and touches both sides of the circle.Geometry · Grade 6, Grade 7, Grade 8SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3PolygonA polygon is a flat, closed shape made of straight line segments, called sides, that meet only at their endpoints. Each corner is a vertex. Polygons are named by their number of sides: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8.Geometry · Grade 2, Grade 3, Grade 5TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5Area of a CircleThe area of a circle is the amount of flat space inside the circle, measured in square units. It equals pi times the radius squared: A = π r2. A circle with radius 5 cm has area 25π, about 78.5 square centimeters. If you know the diameter, halve it first to get the radius.Geometry · Grade 7, Grade 8, GeometrySphereA sphere is a perfectly round solid shape, like a ball, in which every point on the surface is the same distance from the center. That distance is the radius. A sphere has no faces, edges or vertices. Its volume is V = 43π r3 and its surface area is 4π r2.Geometry · Kindergarten, Grade 8, Geometry