Cylinder, Cone and Sphere (Surface Area and Volume)
348. The Cone's Blueprint · h, r, and l in perfect harmony
The slant height l always satisfies l² = h² + r².
In a right circular cone with radius r, height h, and slant height l, the three satisfy l² = h² + r² (Pythagoras). Then curved surface area = πrl, TSA = πr(l + r), and volume = ⅓πr²h.
What this lesson covers
Try to break it
Drag A to change the height h and C to change the base radius r. The slant height ℓ always equals √(h² + r²) — Pythagoras on the right triangle of legs h and r. Pull h tall and ℓ approaches h; spread r wide and ℓ approaches r; in between, ℓ is always larger than both legs.
How you build it
Construct a cone: draw the base radius, mark the apex, drop the perpendicular height, draw the base ellipse, then draw the two slant edges.
- Draw the base radius — a horizontal segment from centre B to rim point C.
- Mark the apex A directly above the centre B.
- From apex A, drop a perpendicular to the base radius. The foot at B is a right angle — this is the height AB.
- Draw the circular base with centre B, passing through rim point C.
- Draw the slant edge AC, joining the apex A to rim point C — this is the slant height l.
- Draw the second slant edge from apex A to the left end of the base ellipse, completing the cone.
The proof, step by step
Prove that the slant height of a cone satisfies l² = h² + r².
- In a right circular cone, the height h, radius r, and slant height l form a right-angled triangle, with the right angle between the height and the radius.
- By the Pythagorean theorem, in a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.
- Therefore, l² = h² + r². This relation is the cone's blueprint.
Worked example
A right circular cone has a base radius of 7 cm and a slant height of 25 cm. Find its total surface area. (Use π = 22/7)
TSA = πr(l+r) = 22/7 × 7 × (25 + 7) = 22 × 32 = 704 cm².
- 616 cm²
- 704 cm² — correct
- 792 cm²
- 880 cm²