Stereometry: Solids of Revolution

Cylinder

It is formed by rotating a rectangle around one of its sides. R is the radius, H is the height.

• Volume: V = πR²H.
• Lateral surface area: lateral area = 2πRH, total surface area 2πR(R + H).
• Axial section: a rectangle 2R × H. If it is a square, H = 2R.

Cone

It is formed by rotating a right triangle around one of its legs. l is the slant height.

Quantity	Formula
Slant height	l² = R² + H²
Volume	V = (1/3) · πR²H
Lateral surface area	lateral area = πRl
Total surface area	S = πR(R + l)

• The axial section is an isosceles triangle (base 2R, lateral side l).
• Frustum of a cone: V = (πH/3) · (R² + Rr + r²), lateral surface area π(R + r)l.

Sphere

• Volume: V = (4/3) · πR³.
• Surface area: S = 4πR².
• If a plane cuts a sphere, the section is a circle. Let the radius of the section be r and the distance from the center to the plane be d: r² = R² − d².
• The tangent plane to the surface of a sphere is perpendicular to the radius.

1. Identify the solid and mark R, H, l on the diagram
2. In a cone, find l or H using the Pythagorean theorem
3. Choose the formula for volume or surface area
4. Keep π in the answer and multiply it out at the end

Examples

• Cylinder R = 3, H = 5: V = 9π · 5 = 45π.
• Cone R = 3, H = 4: l = 5, lateral area π · 3 · 5 = 15π.
• Sphere R = 3: V = (4/3) · π · 27 = 36π.
• Cone R = 6, l = 10: H = 8, V = (1/3) · π · 36 · 8 = 96π.

Mistakes

• The cone's volume uses H, not l. The slant height is used only in the lateral surface area.
• Do not forget the 1/3 in the volume of a cone and a pyramid, and the 4/3 in the volume of a sphere.