A 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.
Every point on a sphere is the same distance from its center. Both formulas need only that one number, the radius.
Formal definition and notation
A sphere with center O and radius r > 0 is the set of all points in three-dimensional space whose distance from O is exactly r. The solid it encloses (often also called a sphere in school) has volumeV = 43π r3, and the sphere itself has surface areaS = 4π r2.
Symbol
Read it as
Meaning
r
r
the radius
d = 2r
d equals 2 r
the diameter, through the center from one side to the other
Every point on the surface is the same distance, r, from the center.
A sphere has one curved surface and no flat faces, edges or vertices.
Volume 43π r3 grows with the cube of the radius: doubling r makes the volume 8 times as big.
Surface area 4π r2 is exactly 4 times the area of a circle with the same radius.
Half a sphere is a hemisphere, with volume 23π r3.
Sphere formulas
Sphere formulas
V = 43π r3
V
volume, in cubic units
r
radius; if you know the diameter, halve it first
S = 4π r2
surface area, in square units
Volume cubes the radius; surface area squares it. Leave answers in terms of π, or use π ≈ 3.14 when the problem asks for a decimal.
Sphere examples, step by step
Example 1: Volume from the radius
Warm-up
A ball has a radius of 3 cm. Find its volume.
V = 43π r3 = 43π (3)3.
33 = 27, and 43 × 27 = 36, so V = 36π.
36 × 3.1416 ≈ 113.1.
Answer: The volume is 36π cubic cm, about 113.1 cubic cm.
Example 2: Surface area from the diameter
Core
A globe has a diameter of 10 inches. What is its surface area?
Halve the diameter: r = 10 ÷ 2 = 5.
S = 4π r2 = 4π (5)2 = 4π × 25 = 100π.
100π ≈ 314.2.
Answer: The surface area is 100π, about 314.2 square inches.
Example 3: A ball in a box
Test-style
A ball with radius 3 inches fits exactly inside a cube-shaped box with 6-inch edges. How much empty space is left in the box? Use π ≈ 3.14.
Box volume: 63 = 216 cubic inches.
Ball volume: 43π (3)3 = 36π ≈ 36 × 3.14 = 113.04.
Empty space: 216 − 113.04 = 102.96.
Answer: About 102.96 cubic inches of the box is empty.
Common mistakes
✗ MistakeA ball with diameter 10: V = 43π (10)3.
✓ InsteadUse the radius: r = 5, so V = 43π (5)3 = 5003π.
The formulas use the radius. Plugging in the diameter makes the volume 8 times too big.
✗ MistakeVolume of a sphere with radius 3: 43π (3)2 = 12π.
✓ InsteadCube the radius for volume: 43π (3)3 = 36π.
Volume is three-dimensional, so the radius is cubed. Squaring is for surface area.
✗ MistakeA circle and a sphere are the same shape.
✓ InsteadA circle is flat (2-D); a sphere is solid (3-D).
A circle has area π r2; a sphere has volume 43π r3 and surface area 4π r2.
Not to be confused with
Spherevs.Circle
A circle is a flat curve where every point is r from the center in a plane; it has area π r2. A sphere is the 3-D version, with volume 43π r3.
Spherevs.Hemisphere
A hemisphere is half a sphere, cut through the center. Its volume is half: 23π r3.
Quick check: 3 questions
Question 1 of 3
What is the volume of a sphere with radius 6?
Show answer and explanation
Answer: B.43π (6)3 = 43 × 216π = 288π. Choice A is the surface area 4π r2, C squares the radius in the volume formula, and D forgets to divide by 3.
Question 2 of 3
What is the surface area of a sphere with radius 2 m?
Show answer and explanation
Answer: A.4π (2)2 = 16π. B is the area of one circle of radius 2, C is the volume, and D doubles instead of squaring.
Question 3 of 3
Which object is shaped most like a sphere?
Show answer and explanation
Answer: B. A basketball is round in every direction. A soup can is a cylinder, a plate is close to a flat circle, and an ice cream cone is a cone.
Sphere: frequently asked questions
What is the difference between a sphere and a circle?
A circle is flat: it lives on a page and has only length and width. A sphere is solid: it has depth too, like a ball. Slice a sphere through its center and the cut edge is a circle with the same radius. You find the area of a circle, but the volume and surface area of a sphere.
Does a sphere have faces, edges or vertices?
No. Faces are flat surfaces, edges are segments where faces meet, and vertices are corners. A sphere has one smooth curved surface and none of those. So you can count faces, edges and vertices on a cube or a pyramid, but not on a sphere. A sphere can still roll in every direction, which no polyhedron can do smoothly.
How do you find the volume of a hemisphere?
Take half the sphere volume: V = 12 × 43π r3 = 23π r3. For a bowl shaped like a hemisphere with radius 6 cm, the volume is 23π × 216 = 144π, about 452.4 cubic cm. Check the answer: it should be exactly half of the full sphere, 288π.
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 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 8DiameterThe 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 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 2CircleA 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 7Area 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, GeometryInchAn inch is a US customary unit of length equal to one twelfth of a foot, about the width of an adult thumb. Its abbreviation is in. Rulers split each inch into halves, quarters, eighths and sixteenths. One inch is exactly 2.54 centimeters, and 12 inches make 1 foot.Measurement · Grade 2, Grade 3, Grade 4SquareA 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 3LengthLength 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 3PyramidA 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 8CircumferenceCircumference 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 8Rectangular PrismA rectangular prism is a solid shape with six rectangular faces, where opposite faces match and every corner is a right angle. A shoebox and a brick are rectangular prisms. It has 12 edges and 8 vertices, and its volume is length times width times height: V = lwh.Geometry · Grade 5, Grade 6, Grade 7CylinderA 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, GeometryConeA 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 8