A 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.
The height h goes straight down inside the cone; the slant height s runs along the outside. Volume uses h, and h, r and s form a right triangle.
Formal definition and notation
A right circular cone is the solid formed by all segments joining a point (the apex) to the points of a disk (the base), where the apex lies on the line perpendicular to the base through its center. The heighth is the distance from the apex to the base, and the slant height s is the distance from the apex to the edge of the base, with s2 = r2 + h2.
Symbol
Read it as
Meaning
h
h, the height
straight down from the apex to the center of the base, at a right angle
s or ℓ
slant height
along the side, from the apex to the edge of the base
Key facts
A cone has 1 flat circular face, 1 curved surface and 1 vertex (the apex).
Three cones of water exactly fill a cylinder with the same base and height, which is where the 13 comes from.
The height, the radius and the slant height form a right triangle, with the slant height as the hypotenuse.
Spinning a right triangle around one of its legs sweeps out a cone.
A cone and a pyramid share the volume rule V = 13 × base area × h.
Cone formulas
Cone formulas
V = 13π r2h and SA = π r2 + π rs
r
radius of the base
h
perpendicular height (use for volume)
s
slant height √r2 + h2 (use for the curved surface)
π rs
lateral area: the curved side only
Use h for volume and s for surface area. If you are given one, the Pythagorean theorem finds the other.
Cone examples, step by step
Example 1: Volume of a cone
Warm-up
A cone has radius 3 cm and height 8 cm. Find its volume in terms of π.
V = 13π r2h.
r2h = 9 × 8 = 72.
V = 13 × 72π = 24π.
Answer:V = 24π ≈ 75.4cm3.
Example 2: Slant height from the height
Core
A cone has radius 5 in and height 12 in. Find its slant height and its total surface area.
s = √r2 + h2 = √25 + 144 = √169 = 13.
Base area: π(5)2 = 25π.
Curved side: π rs = π(5)(13) = 65π.
Total: 25π + 65π = 90π ≈ 282.7.
Answer:s = 13 in, and the surface area is 90π ≈ 282.7in2.
Example 3: Cone vs cylinder
Test-style
A cylindrical glass and a conical cup have the same radius, 4 cm, and the same height, 9 cm. How many full cups does it take to fill the glass, and what is the cup’s volume?
Glass: π(4)2(9) = 144π.
Cup: 13 × 144π = 48π.
144π ÷ 48π = 3.
Answer: 3 cups fill the glass; each cup holds 48π ≈ 150.8cm3.
Common mistakes
✗ MistakeForgetting the 13: r = 3, h = 4 gives V = 36π.
✓ InsteadV = 13π(9)(4) = 12π.
36π is the cylinder with the same base and height. A cone holds only a third of that.
✗ MistakeUsing the slant height in the volume formula: r = 6, s = 10 gives 13π(36)(10) = 120π.
✓ InsteadFind h = √102 − 62 = 8, so V = 13π(36)(8) = 96π.
Volume needs the perpendicular height, which is shorter than the slant height.
✗ MistakeLateral area = π rh.
✓ InsteadLateral area = π rs, with the slant height: for r = 6, s = 10, it is 60π.
The curved surface stretches along the slant, so its area uses s, not h.
Not to be confused with
Conevs.Pyramid
A pyramid has a polygon base and flat triangular sides; a cone has a circular base and one curved side. Both have volume 13 × base area × h.
A cylinder has two circular bases and no apex. With the same base and height, a cylinder holds 3 times what a cone holds.
Quick check: 3 questions
Question 1 of 3
What is the volume of a cone with radius 6 ft and height 5 ft?
Show answer and explanation
Answer: B.13π(36)(5) = 60π. 180π forgets the 13, 10π forgets to square the radius, and 20π uses r = 2.
Question 2 of 3
A cone has radius 9 and slant height 15. What is its height?
Show answer and explanation
Answer: A.h = √152 − 92 = √144 = 12. √306 adds the squares instead of subtracting, 6 subtracts the lengths, and 24 adds them.
Question 3 of 3
A cone and a cylinder have the same base and the same height. The cylinder holds 270 mL. How much does the cone hold?
Show answer and explanation
Answer: C. A cone holds one third as much: 270 ÷ 3 = 90 mL. 810 multiplies by 3, 135 halves, and 270 assumes they are equal.
Cone: frequently asked questions
Why is the volume of a cone one third of a cylinder?
Fill a cone with water and pour it into a cylinder with the same base and height: it takes exactly three cones to fill it. Calculus proves the same thing by adding up thin circular slices whose radius shrinks steadily to zero at the apex. Pyramids follow the same one-third rule compared with prisms of the same base and height.
What is the difference between height and slant height?
The height h runs straight down from the apex to the center of the base, inside the cone, at a right angle to the base. The slant height s runs along the outside surface from the apex to the edge of the base. Since they form a right triangle with the radius, s2 = r2 + h2, so s is always longer than h.
VolumeVolume 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 7RadiusThe 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 8HeightHeight 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 6Right AngleA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Geometry · Grade 2, Grade 3, Grade 4SideA 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 2CylinderA 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 radius r of a base and the height h between the bases. Its volume is π r2h, the base area times the height.Geometry · Grade 1, Grade 8, GeometryRight TriangleA right triangle is a triangle with one right angle, an angle of exactly 90 degrees. The two sides that form the right angle are the legs, and the longest side, across from the right angle, is the hypotenuse. Its sides follow the Pythagorean theorem: a2 + b2 = c2.Geometry · Grade 4, Grade 6, Grade 8HypotenuseThe hypotenuse is the longest side of a right triangle. It is always across from the right angle, and the two shorter sides that form the right angle are the legs. In the Pythagorean theorem a2 + b2 = c2, the hypotenuse is c: in a 3-4-5 right triangle, it is the side of length 5.Geometry · Grade 8, Algebra 1PyramidA pyramid is a solid figure with one polygon as its base and triangular faces that rise from the base edges and meet at a single top point, the apex. A pyramid is named by its base, such as a square pyramid. Its volume is one third of the base area times the height: V = 13Bh.Geometry · Grade 6, Grade 7, Grade 8Pythagorean TheoremThe Pythagorean theorem says that in a right triangle, a2 + b2 = c2. The legs are a and b, the two sides that form the right angle. The hypotenuse, c, is the longest side across from the right angle. Legs of 3 and 4 give 9 + 16 = 25, so the hypotenuse is 5.Geometry · Grade 8, Algebra 1Surface 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 8PolygonA 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 5CircleA 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 7SphereA 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