Pyramids and Cones
Pyramids and cones hold one third of the matching prism or cylinder.
About this topic
A pyramid has a polygon base and triangular faces meeting at an apex, and a cone has a circular base tapering to a point. Their volume is one-third of the matching prism or cylinder: V = ⅓Bh. A cone's volume is ⅓πr²h.
Surface area includes the base plus lateral faces, using the slant height. For a cone, lateral area is πrl, where l is the slant height found from the Pythagorean theorem. Pyramids and cones appear in architecture, funnels and tents.
Worked example: a cone has radius 3 and height 4, so the slant height is 5 and the volume is ⅓π(9)(4) = 12π ≈ 37.7. Its lateral area is π(3)(5) = 15π, and with the base 9π, the total surface area is 24π. A square pyramid with a base of side 6 and height 4 has volume ⅓(36)(4) = 48.
Ice cream cones, the Great Pyramid and traffic cones show these shapes. Students often forget the one third, or use the slant height in the volume formula where the vertical height belongs. The slant height is used for the sides' area only. Three identical cones of water fill one cylinder of the same base and height, a good way to remember. Use the Pythagorean theorem to move between height, radius and slant height.
Key ideas
- V = ⅓ Bh
- Cone V = ⅓πr²h
- Slant height differs from height
Quick check
Use cases: where Pyramids and Cones helps
- Homework: review the key idea behind Pyramids and Cones before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Pyramids and Cones without notes.
- Visual learning: play with the interactive lab to see how Pyramids and Cones changes when you adjust its inputs.
- Teaching a friend: explain Pyramids and Cones in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Pyramids and Cones?
Pyramids and cones hold one third of the matching prism or cylinder.
What will I learn in Pyramids and Cones?
V = ⅓ Bh. Cone V = ⅓πr²h. Slant height differs from height.
Is Pyramids and Cones part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Pyramids and Cones lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Pyramids and Cones?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
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