A 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.
The height drops straight down inside the pyramid; the slant height runs along a face. Volume uses the height: V = 13Bh.
Formal definition and notation
A pyramid is a polyhedron formed by a polygonal base and a point (the apex) not in the plane of the base, together with the triangles that join the apex to each side of the base. Its heighth is the perpendicular distance from the apex to the plane of the base, and its volume is V = 13Bh, where B is the base area.
A pyramid has a rectangular base 8 cm by 5 cm and a volume of 160 cubic centimeters. What is its height?
Base area: B = 8 × 5 = 40.
Set up 160 = 13 × 40 × h.
Multiply both sides by 3: 480 = 40h.
Divide by 40: h = 12.
Answer: The height is 12 cm.
Common mistakes
✗ MistakeVolume of a pyramid with base area 36 and height 10: 36 × 10 = 360.
✓ InsteadMultiply by one third: 13 × 36 × 10 = 120.
Bh is the volume of a prism with the same base and height. A pyramid fills only one third of it.
✗ MistakeUsing the slant height (the height of a face) as h in the volume formula.
✓ InsteadUse the perpendicular height from the apex straight down to the base.
Volume depends on how high the apex sits above the base. The slant height is longer and gives too big an answer.
✗ MistakeFor a square base of side 6, using B = 6.
✓ InsteadB is the base area: B = 62 = 36.
B stands for an area in square units. Using a side length leaves the answer in the wrong units and far too small.
Not to be confused with
Pyramidvs.Prism
A prism has two matching bases joined by rectangles; a pyramid has one base and triangles meeting at a point. With the same base and height, Vprism = Bh and Vpyramid = 13Bh.
Pyramidvs.Cone
A cone has a circular base and a curved side; a pyramid has a polygon base and flat triangular faces. Both use one third of base area times height.
Quick check: 3 questions
Question 1 of 3
A triangular pyramid has a base area of 15 square inches and a height of 8 inches. What is its volume?
Show answer and explanation
Answer: D.V = 13 × 15 × 8 = 40. Choice A forgets the one third, B takes one half (the triangle-area habit), and C adds the numbers.
Question 2 of 3
How many faces does a square pyramid have?
Show answer and explanation
Answer: B. One square base plus 4 triangular faces makes 5. Choice A forgets the base, C is a cube, and 8 is the number of edges.
Question 3 of 3
In V = 13Bh for a pyramid, what is h?
Show answer and explanation
Answer: C. The height is measured straight down from the apex, at a right angle to the base. The slanted edge and the face height (slant height) are both longer, and the perimeter is not a height at all.
Pyramid: frequently asked questions
Why is the volume of a pyramid one third of base times height?
Three pyramids with the same base and height fill a prism exactly. You can see it by pouring: a pyramid full of sand fills a matching prism after 3 pours. For a cube, three congruent pyramids that share one corner fit together to make the whole cube, so each holds one third.
What is the difference between a pyramid and a prism?
A prism has two identical, parallel bases joined by rectangular faces, like a box. A pyramid has a single base, and its other faces are triangles that meet at one point, the apex. Prism volume is base area times height; pyramid volume is one third of that.
Is a cone a pyramid?
No. A pyramid must have a polygon for its base and flat triangular faces. A cone has a circular base and one curved surface. The two are related, though: both come to a point, and both have volume equal to one third of base area times height.
PolygonA 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 5SideA 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 2HeightHeight 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 6VolumeVolume 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 7NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2SquareA 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 3Surface 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 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 8CircleA 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 7MeterA meter (m) is a metric unit of length equal to 100 centimeters. It is the base unit of length in the metric system, a little over three feet (about 3.28 feet). A doorway is about 2 meters tall, and 1,000 meters make a kilometer. To change meters to centimeters, multiply by 100.Measurement · Grade 3, Grade 4, Grade 5CentimeterA centimeter is a metric unit of length equal to one hundredth of a meter, about the width of a fingernail. Its symbol is cm. There are 100 centimeters in 1 meter and 10 millimeters in 1 centimeter, and 1 inch equals 2.54 centimeters.Measurement · Grade 3, Grade 4, Grade 5AreaArea is the amount of surface a flat shape covers, measured in square units. It counts how many unit squares fit inside the shape, which is why the units are squared. A rectangle’s area is length times width, so a 5 by 3 rectangle has area 15 square units.Geometry · Grade 3, Grade 4, Grade 5Rectangular 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 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