Prism vs Pyramid — Difference (with Table & FAQs)
Prism vs Pyramid compared side by side — a 5-row comparison table, 4 key points and 7 exam FAQs. Class 7-9 Maths, free and no sign-up.
TL;DR: Prism vs Pyramid compared side by side — a 5-row comparison table, 4 key points and 7 exam FAQs. Class 7-9 Maths, free and no sign-up.
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
The main difference between Prism and Pyramid: A prism is a 3D shape with two parallel, congruent polygonal bases and rectangular sides. A pyramid is a 3D shape with one polygonal base and triangular si
Prism vs Pyramid — Comparison Table
| Basis | Prism | Pyramid |
|---|---|---|
| Bases | Two parallel, congruent polygonal bases | One polygonal base |
| Sides | Rectangular or parallelogram sides | Triangular sides |
| Apex | No apex, has parallel edges | Single apex where triangles meet |
| Vertices | More vertices due to two bases | Fewer vertices, one at apex |
| Examples | Cube, cuboid, triangular prism | Tetrahedron, square pyramid, cone |
Key Points
- A prism has two parallel bases connected by rectangular faces
- A pyramid has one base with triangular faces meeting at an apex
- Prisms have uniform cross-sections while pyramids taper to a point
- Both are important 3D shapes in geometry and architecture
A prism has two identical bases
A prism is a solid with two identical, parallel polygonal faces called bases, joined by flat side faces that are always rectangles (or parallelograms). Slice a prism anywhere parallel to its base and you get exactly the same shape every time — the cross-section is uniform along its whole length.
Prisms are named after the shape of the base. A triangular prism has two triangular bases, a pentagonal prism has two pentagonal ones. A cuboid is a rectangular prism, and a cube is a square prism, which surprises most students the first time they meet it.
A pyramid has one base and a point
A pyramid has just one polygonal base, and its side faces are triangles that all meet at a single point called the apex. The cross-section is not uniform: slice a pyramid parallel to its base and the slices get smaller and smaller as you approach the apex.
Pyramids are also named after the base. A triangular pyramid — also called a tetrahedron — has a triangular base and three triangular sides. A square pyramid, like the pyramids at Giza, has a square base and four triangular sides.
Faces, edges and vertices — the numbers to know
For a base with n sides, both solids follow a simple rule. Learning the rule beats memorising individual cases.
- Prism with an n-sided base: n + 2 faces, 3n edges, 2n vertices. A triangular prism therefore has 5 faces, 9 edges and 6 vertices.
- Pyramid with an n-sided base: n + 1 faces, 2n edges, n + 1 vertices. A square pyramid therefore has 5 faces, 8 edges and 5 vertices.
- Both satisfy Euler's formula, F + V − E = 2. Check the triangular prism: 5 + 6 − 9 = 2. Check the square pyramid: 5 + 5 − 8 = 2.
- Euler's formula is a fast way to verify an answer in the exam — if your three numbers do not give 2, one of them is wrong.
The volume formulas, and the one-third that matters
Volume of a prism = base area × height. Volume of a pyramid = ⅓ × base area × height.
That factor of one-third is the most examined single fact in this topic. It means a pyramid holds exactly one third of the prism that shares its base and height — three identical pyramids can be fitted together to fill their corresponding prism exactly.
The lateral surface of a prism is made of rectangles, so lateral surface area = perimeter of base × height. The lateral surface of a pyramid is made of triangles, so lateral surface area = ½ × perimeter of base × slant height. Note that a pyramid's formula uses the slant height, not the vertical height — mixing the two is a guaranteed lost mark.
Telling them apart in one glance
Count the bases. Two identical parallel bases means a prism; one base and an apex means a pyramid.
Or look at the side faces. Rectangles all round means a prism; triangles meeting at a point means a pyramid. Either test works instantly on a diagram.
Prism vs Pyramid — Worked Examples
- A Toblerone-style chocolate box — A triangular prism — two identical triangular ends joined by three rectangles.
- The Great Pyramid of Giza — A square pyramid — one square base and four triangular faces meeting at an apex.
- A brick — A rectangular prism (a cuboid). Its cross-section is the same all the way along.
- Volume comparison — A prism and a pyramid both with base area 12 cm² and height 5 cm: the prism holds 12 × 5 = 60 cm³, the pyramid ⅓ × 12 × 5 = 20 cm³, exactly one third.
Common Mistakes Students Make
- Forgetting the ⅓ in the volume of a pyramid. This is the most frequently lost mark in the topic.
- Using the vertical height instead of the slant height when finding a pyramid's lateral surface area.
- Not realising that a cube and a cuboid are prisms.
- Counting a square pyramid as having 4 faces. It has 5 — four triangles plus the base.
- Assuming the side faces of a prism must be squares. They are rectangles, and only in special cases squares.
Diagrams
Where This Shows Up in Real Life
Why water tanks are prisms and roofs are pyramids
A storage tank is built as a prism because a uniform cross-section means the volume is simply base area times height, and the water level rises evenly and predictably as it fills. A roof is often pyramidal because sloping triangular faces shed rain and snow and meet at a single ridge or apex. The shape follows the job: uniform cross-section for holding, converging faces for shedding.
The Great Pyramid and the one-third rule
The pyramids at Giza are square pyramids, and their volume is one third of the prism that would share the same base and height. That factor is not a convention — three identical pyramids can be fitted together to fill their corresponding prism exactly. It is why a pyramid of the same footprint and height needs only a third of the material, and why the shape is so stable: most of the mass sits low.
Packaging design
A Toblerone box is a triangular prism, chosen because a prism stacks efficiently and its uniform cross-section makes filling machinery simple. Pyramid-shaped cartons exist but are rare, precisely because they waste space when stacked. Comparing surface area to volume is how a designer decides: less packaging material for the same contents means lower cost.
Euler’s formula as an exam safety net
For any convex polyhedron, F + V − E = 2. If you have counted faces, vertices and edges and the arithmetic does not give 2, one of your three numbers is wrong. It costs five seconds and catches the miscounts that lose whole marks — for instance forgetting that a square pyramid has five faces, not four, because the base counts.
Solved Numericals — Step by Step
Problem 1. A triangular prism has a base area of 24 cm² and a height of 10 cm. Find its volume. A pyramid has the same base and height — find its volume.
- Volume of a prism = base area × height = 24 × 10 = 240 cm³.
- Volume of a pyramid = ⅓ × base area × height.
- = ⅓ × 24 × 10 = 240 ÷ 3.
Answer: Prism 240 cm³, pyramid 80 cm³ — exactly one third, as the formula requires.
Problem 2. A square pyramid has a base edge of 6 cm and a slant height of 5 cm. Find its lateral surface area and total surface area.
- Perimeter of the square base = 4 × 6 = 24 cm.
- Lateral surface area = ½ × perimeter × slant height = ½ × 24 × 5 = 60 cm².
- Base area = 6² = 36 cm².
- Total surface area = 60 + 36 = 96 cm².
Answer: Lateral 60 cm², total 96 cm². Note the SLANT height is used here, not the vertical height.
Problem 3. Verify Euler’s formula for a pentagonal prism.
- For an n-sided base with n = 5: faces = n + 2 = 7.
- Edges = 3n = 15. Vertices = 2n = 10.
- Euler: F + V − E = 7 + 10 − 15.
Answer: 2. The formula holds, confirming the counts of 7 faces, 15 edges and 10 vertices.
Board-Style Questions with Model Answers
Distinguish between a prism and a pyramid. (3 marks)
A prism is a solid with two identical and parallel polygonal faces called bases, joined by lateral faces that are rectangles or parallelograms. Its cross-section taken parallel to the base is the same all along its length, and its volume is base area multiplied by height. A pyramid has only one polygonal base, and its lateral faces are triangles meeting at a single point called the apex. Its cross-section decreases steadily towards the apex, and its volume is one third of base area multiplied by height. Both are named after the shape of the base, so a triangular prism has triangular bases while a square pyramid has a square base. The quickest test on a diagram is to count the bases: two means a prism, one plus an apex means a pyramid.
State the number of faces, edges and vertices for a prism and a pyramid with an n-sided base, and verify Euler’s formula. (3 marks)
A prism with an n-sided base has n + 2 faces, 3n edges and 2n vertices. Substituting into Euler’s formula gives F + V − E = (n + 2) + 2n − 3n = n + 2 + 2n − 3n = 2, which holds for every value of n. A pyramid with an n-sided base has n + 1 faces, 2n edges and n + 1 vertices. Substituting gives F + V − E = (n + 1) + (n + 1) − 2n = 2n + 2 − 2n = 2, which again holds for every n. For example, a triangular prism has 5 faces, 9 edges and 6 vertices, giving 5 + 6 − 9 = 2, and a square pyramid has 5 faces, 8 edges and 5 vertices, giving 5 + 5 − 8 = 2.
Why is the volume of a pyramid one third that of a prism with the same base and height? (2 marks)
Because three identical pyramids having the same base and the same height can be fitted together to fill exactly the volume of the corresponding prism, leaving no gaps and no overlap. This geometric fact, which can be demonstrated physically with three hollow models and a quantity of sand or water, is the origin of the factor of one third in the formula. It is not an approximation or a convention: the relationship is exact, so a pyramid always holds precisely one third of the prism sharing its base and height.
A cube is sometimes described as a prism. Justify this statement. (2 marks)
A prism is defined as a solid with two identical parallel polygonal bases joined by rectangular lateral faces, with a uniform cross-section along its length. A cube satisfies every part of this definition: it has two identical parallel square faces acting as bases, the four remaining faces joining them are rectangles which in this special case happen to be squares, and any slice taken parallel to the base is an identical square. A cube is therefore a square prism, and a cuboid is likewise a rectangular prism. This is why the volume of a cube, a³, is simply base area a² multiplied by height a — the general prism formula applied to a special case.
Assertion–Reason Questions (CBSE Format)
For each question choose: (a) both A and R are true and R is the correct explanation of A; (b) both are true but R is not the correct explanation; (c) A is true but R is false; (d) A is false but R is true.
1. Assertion (A): The volume of a pyramid is one third that of a prism with the same base and height.
Reason (R): Three such pyramids can be assembled to fill the corresponding prism exactly.
Answer: (a) — Both are true and the reason is the geometric justification for the assertion.
2. Assertion (A): A square pyramid has four faces.
Reason (R): Its lateral faces are four triangles meeting at an apex.
Answer: (d) — The assertion is false — it has five faces, because the square base counts as one. The reason correctly describes the four lateral faces and is exactly what tempts students to forget the base.
3. Assertion (A): A cube is a prism.
Reason (R): It has two identical parallel bases and a uniform cross-section.
Answer: (a) — Both are true and the reason satisfies the definition of a prism. A cube is a square prism.
4. Assertion (A): Euler’s formula can be used to check a count of faces, edges and vertices.
Reason (R): For any convex polyhedron, F + V − E = 2.
Answer: (a) — Both are true and the reason explains the assertion. If the arithmetic does not give 2, one of the three counts is wrong.
Quick Self-Check
Q1. How many faces does a square pyramid have?
- A. 4
- B. 5
- C. 6
- D. 8
Answer: B. 5 — Four triangular side faces plus the square base makes 5. The formula n + 1 gives 4 + 1 = 5.
Q2. The volume of a pyramid is what fraction of a prism with the same base and height?
- A. One half
- B. One third
- C. Two thirds
- D. The same
Answer: B. One third — Volume of a pyramid = ⅓ × base area × height, so it is exactly one third of the corresponding prism.
Q3. A triangular prism has how many edges?
- A. 6
- B. 9
- C. 12
- D. 5
Answer: B. 9 — For an n-sided base a prism has 3n edges, so 3 × 3 = 9. Check with Euler: 5 + 6 − 9 = 2.
Q4. Which solid has two identical parallel bases?
- A. Pyramid
- B. Cone
- C. Prism
- D. Sphere
Answer: C. Prism — A prism is defined by two identical, parallel polygonal bases joined by rectangular side faces.
Key Terms Glossary
- Prism
- A solid with two identical parallel polygonal bases joined by rectangular lateral faces. Uniform cross-section.
- Pyramid
- A solid with one polygonal base and triangular lateral faces meeting at an apex.
- Apex
- The single point where all the triangular faces of a pyramid meet.
- Base
- The polygonal face a solid is named after. A prism has two; a pyramid has one.
- Lateral faces
- The side faces excluding the base(s). Rectangles in a prism, triangles in a pyramid.
- Cross-section
- The shape revealed by slicing a solid. Uniform in a prism, shrinking towards the apex in a pyramid.
- Slant height
- The distance from the apex down the middle of a triangular face to the base edge. Used for lateral surface area.
- Height (altitude)
- The perpendicular distance from the apex or top base straight down to the base. Used for volume.
- Euler’s formula
- F + V − E = 2 for any convex polyhedron. A fast way to check your counts.
- Tetrahedron
- A triangular pyramid — 4 faces, 6 edges, 4 vertices.
- Cuboid
- A rectangular prism; a box whose length, breadth and height may differ.
- Polyhedron
- A solid bounded entirely by flat polygonal faces.
One-Minute Revision
- PRISM = TWO identical parallel bases, RECTANGULAR sides, uniform cross-section.
- PYRAMID = ONE base + an APEX, TRIANGULAR sides, cross-section shrinks upward.
- Volume: prism = base area × height. Pyramid = ⅓ × base area × height. The ⅓ is the most-lost mark.
- Prism with n-sided base: n+2 faces, 3n edges, 2n vertices.
- Pyramid with n-sided base: n+1 faces, 2n edges, n+1 vertices.
- A square pyramid has FIVE faces — the base counts. Triangular prism: 5 faces, 9 edges, 6 vertices.
- Lateral surface area of a pyramid uses the SLANT height; volume uses the VERTICAL height.
- Euler check: F + V − E = 2. If it does not, one of your counts is wrong.
Syllabus reference: NCERT Class 8 Maths, Chapter 10 — Visualising Solid Shapes; surface areas and volumes in Class 9 and Class 10.
Frequently Asked Questions
What is the difference between a prism and a pyramid?
A prism has two identical parallel bases joined by rectangular faces (e.g. a triangular prism). A pyramid has just one base, with triangular faces that meet at a single top point called the apex.
How many bases does a prism and a pyramid have?
A prism has two identical bases; a pyramid has only one base.
Give an everyday example of a prism and a pyramid.
Prism: a Toblerone chocolate box (triangular prism). Pyramid: the pyramids of Egypt (square pyramid).
Is a cube a prism or a pyramid?
A cube is a prism — specifically a square prism. It has two identical parallel square bases joined by rectangular (in this case square) side faces, and its cross-section is the same all the way along.
What is Euler's formula and how do I use it?
Euler's formula states that F + V − E = 2, where F is the number of faces, V the vertices and E the edges of any convex polyhedron. Use it to check your answer: a square pyramid gives 5 + 5 − 8 = 2, so the counts are right.
Why is a pyramid's volume one third of a prism's?
Because three identical pyramids with the same base and height can be fitted together to fill the corresponding prism exactly. That geometric fact is where the ⅓ in the formula comes from.
What is the difference between the height and the slant height of a pyramid?
The height is the perpendicular distance from the apex straight down to the centre of the base. The slant height is the distance from the apex down the middle of a triangular side face to the base edge. Volume uses the height; lateral surface area uses the slant height.