Find the surface area and volume of a pyramid or cone using slant height and the one–third relationship to prisms and cylinders.
Tapering to an apex
A pyramid and a cone are what you get when a prism or cylinder's top base shrinks to a single point — the apex. That taper changes both formulas: the lateral faces are no longer rectangles but triangles (or a curved wedge, for a cone), and the solid encloses far less space than a prism or cylinder with the same base and height.
Slant height vs. height
Don't confuse the two: the height (h) is the perpendicular distance from the apex straight down to the base. The slant height (l) runs along a lateral face, from the apex down to the midpoint of a base edge (pyramid) or to the base circle (cone). Slant height is always longer than height — the two, together with half the base's width, form a right triangle.
Surface area and volume formulas
Let B be the base area, P the base perimeter, l the slant height, h the height, and r the radius of a circular base.
Solid
Surface area
Volume
Pyramid
S.A. = B + ½Pl
V = ⅓Bh
Cone
S.A. = πr² + πrl
V = ⅓πr²h
The ⅓ in both volume formulas isn't a coincidence: a pyramid or cone always holds exactly one–third the volume of a prism or cylinder that shares the same base and height — three copies of the pointed solid pack perfectly into the "full" one.
In a cone (or pyramid), height h, slant height l, and the base radius r form a right triangle.
Worked Example 1 · Square pyramid
ProblemA square pyramid has a base edge of 10 m and a slant height of 13 m. Find its surface area.
1.Base area: B = 10² = 100 m². Base perimeter: P = 4(10) = 40 m.
2.S.A. = B + ½Pl = 100 + ½(40)(13) = 100 + 260 = 360 m².
Surface area = 360 m²
Worked Example 2 · Cone volume
ProblemA cone has radius 6 cm and height 15 cm. Find its volume in terms of π, then to the nearest tenth.
1.V = ⅓πr²h = ⅓π(6)²(15).
2.= ⅓π(36)(15) = ⅓(540π) = 180π.
3.180π ≈ 565.5 cm³.
Volume = 180π cm³ ≈ 565.5 cm³
Guided practice
1.A cone has radius 5 in and height 12 in. Find its slant height.Show answer
l² = r² + h² = 5² + 12² = 25 + 144 = 169, so l = 13 in.
2.Using the cone from Problem 1, find its surface area in terms of π.Show answer
3.A square pyramid has base edge 8 ft and height 3 ft. Find its volume.Show answer
B = 8² = 64 ft². V = ⅓Bh = ⅓(64)(3) = 64 ft³.
4.A prism and a pyramid share the same base and the same height. The prism's volume is 90 cm³. What is the pyramid's volume?Show answer
A pyramid always has ⅓ the volume of a prism with the same base and height, so the pyramid's volume is ⅓(90) = 30 cm³.
5.Why is slant height always greater than height for a cone (with r > 0)?Show answer
Slant height is the hypotenuse of a right triangle with legs r and h (l² = r² + h²). A hypotenuse is always longer than either leg, so as long as r > 0, l must be greater than h.