Pyramids and cones taper from a base to one vertex. That taper creates the factor 1/3 in volume. Surface-area questions use lateral faces or a curved lateral surface, so they often require slant height instead of perpendicular height.
Perpendicular height versus slant height
Right-pyramid anatomyA right square pyramid has base edge 12 units, perpendicular height 8 units from the vertex to the base center, and lateral-face slant height 10 units.Right-pyramid anatomyA right square pyramid has base edge 12 units, perpendicular height 8 units from the vertex to the base center, and lateral-face slant height 10 units.
Figure not drawn to scale.
The two heights serve different purposes
Measurement
Location
Used for
Perpendicular height h
Vertex to the base plane at 90°
Volume
Slant height ℓ
Along a lateral face from its base edge to the vertex
Lateral and total surface area
Pyramid formulas
Pyramid
A polyhedron with one polygonal base and triangular lateral faces that meet at a common vertex.
Square-pyramid netA square-pyramid net contains one square base and four triangular lateral faces; each triangular altitude is slant height l.Square-pyramid netA square-pyramid net contains one square base and four triangular lateral faces; each triangular altitude is slant height l.
Right-pyramid formulas
Quantity
Formula
Meaning
Lateral area
\(LA=\tfrac12Pℓ\)
Area of the triangular lateral faces
Total area
\(SA=\tfrac12Pℓ+B\)
Lateral area plus one base
Volume
\(V=\tfrac13Bh\)
One third of a matching prism
Pyramid lateral and total area
\[LA=\frac12P\ell,\qquad SA=\frac12P\ell+B\]
P is base perimeter, B is base area, and ℓ is lateral-face slant height.
Pyramid volume
\[V=\frac13Bh\]
h is perpendicular to the base plane.
Prism and pyramid with matching base and heightA prism and a pyramid have the same base area B and perpendicular height h; the pyramid occupies exactly one third of the prism's volume.Prism and pyramid with matching base and heightA prism and a pyramid have the same base area B and perpendicular height h; the pyramid occupies exactly one third of the prism's volume.Worked example
Square-pyramid area and volume
A right square pyramid has base edge 12 cm, perpendicular height 8 cm, and slant height 10 cm. Find its total surface area and volume.
Base data
B=12²=144 cm² and P=4(12)=48 cm.
Surface area
SA=(1/2)(48)(10)+144=384 cm².
Volume
V=(1/3)(144)(8)=384 cm³.
The total surface area is 384 cm² and the volume is 384 cm³.
Cone formulas
Right-cone anatomyA right cone has radius 6 units, perpendicular height 8 units, and slant height 10 units; the three lengths form a right triangle through the cone's axis.Right-cone anatomyA right cone has radius 6 units, perpendicular height 8 units, and slant height 10 units; the three lengths form a right triangle through the cone's axis.
Figure not drawn to scale.
Right circular cone
A solid with one circular base and a vertex directly above the base center; its perpendicular height, radius, and slant height form a right triangle.
Right-cone formulas
Quantity
Formula
What is included
Lateral area
\(LA=πrℓ\)
Curved surface only
Total area
\(SA=πrℓ+πr²\)
Curved surface and circular base
Volume
\(V=\tfrac13πr²h\)
Space inside the cone
Cone surface area
\[LA=\pi r\ell,\qquad SA=\pi r\ell+\pi r^2\]
Use ℓ for the curved lateral surface and add πr² only if the base is exposed.
Cone volume
\[V=\frac13\pi r^2h\]
Use perpendicular height h, not slant height ℓ.
Slant-height relationship
\[\ell^2=r^2+h^2\]
This follows from the right triangle in an axial cross-section of a right cone.
Worked example
Recover slant height before surface area
A right cone has radius 6 m and height 8 m. Find its total surface area.
Find ℓ
ℓ=√(6²+8²)=10 m.
Lateral area
πrℓ=π(6)(10)=60π m².
Add the base
SA=60π+π(6²)=96π m².
The total surface area is 96π m².
Circumference, rates, and reverse information
Translate before substituting
Convert the given
From circumference C, use r=C/(2π).
Recover a missing length
Use ℓ²=r²+h² when two of r, h, and ℓ are known.
Choose a quantity
Area uses ℓ; volume and filling rates use h.
Check units
Area is square units; volume and volume rate are cubic units.
Parallel cuts and similar cones
Cone cut parallel to its baseA plane parallel to a cone's base removes a smaller similar cone of radius 3 and height 6 from a full cone of radius 9 and height 18, leaving a frustum of height 12.Cone cut parallel to its baseA plane parallel to a cone's base removes a smaller similar cone of radius 3 and height 6 from a full cone of radius 9 and height 18, leaving a frustum of height 12.
Use the full and removed cones' own radii and perpendicular heights.
Worked example
Parallel cut
A cone has radius 9 and height 18. A parallel cut removes the top cone of height 6. Find the remaining volume.
Similarity
The linear scale is 6/18=1/3, so the small radius is 3.
Full cone
V=(1/3)π(9²)(18)=486π.
Small cone
V=(1/3)π(3²)(6)=18π.
Subtract
486π−18π=468π.
The frustum volume is 468π cubic units.
Composite and inscribed solids
Two cones inside a cylinderTwo congruent cones share a vertex at the midpoint of a cylinder. Each cone has radius 5 and height 7, while the cylinder has radius 5 and total height 14.Two cones inside a cylinderTwo congruent cones share a vertex at the midpoint of a cylinder. Each cone has radius 5 and height 7, while the cylinder has radius 5 and total height 14.
Figure not drawn to scale.
Same-base, same-height volume relationships
Prism and pyramid
The pyramid has one third of the prism's volume.
Cylinder and cone
The cone has one third of the cylinder's volume.
Two congruent cones in a matching cylinder
Together the cones occupy one third of the cylinder when each cone's height is half the cylinder height; two thirds remains empty.
Chapter 20 formula reference
Surface area and volume of the six core solids
Solid
Lateral or surface area
Total area
Volume
Right prism
\(Ph\)
\(Ph+2B\)
\(Bh\)
Cube
four lateral faces: \(4s²\)
\(6s²\)
\(s³\)
Right cylinder
\(2πrh\)
\(2πrh+2πr²\)
\(πr²h\)
Sphere
\(4πr²\)
same
\(\tfrac43πr³\)
Right pyramid
\(\tfrac12Pℓ\)
\(\tfrac12Pℓ+B\)
\(\tfrac13Bh\)
Right cone
\(πrℓ\)
\(πrℓ+πr²\)
\(\tfrac13πr²h\)
Mini check
Cone check
A cone has diameter 10 cm, height 12 cm, and slant height 13 cm. What is its total surface area?
65π cm²
90π cm²
120π cm²
300π cm²
Show answer and explanation
Answer: 90π cm².
The radius is 5. SA=πrℓ+πr²=π(5)(13)+π(5²)=65π+25π=90π cm².
Key takeaways
Pyramid and cone essentials
Pyramid volume is (1/3)Bh; cone volume is (1/3)πr²h.
Surface area uses slant height ℓ, while volume uses perpendicular height h.
A right cone satisfies ℓ²=r²+h².
Parallel cuts create similar solids, so linear, area, and volume factors are k, k², and k³.
Composite problems add occupied volumes or subtract them from a containing solid.
Continue learning
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.