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Geometry Unit 8: Surface area & volume

Volume measures space inside a 3D solid; surface area measures the outside. Most volume formulas are base area times height, sometimes with a ⅓.

Covers: Prisms and cylinders, pyramids and cones, spheres, surface area, scaling solids

Study guide

Solids are three-dimensional figures. Prisms and cylinders have two congruent parallel bases, and their volume is base area × height: V = Bh, so a cylinder is πr²h. Pyramids and cones come to a point, and their volume is one third of the matching prism or cylinder: V = ⅓Bh, so a cone is ⅓πr²h. A sphere's volume is (4/3)πr³. Cavalieri's principle says solids with the same height and equal cross-sectional areas at every level have the same volume, which is why an oblique cylinder has the same volume as a right one.

Surface area is the total area of all faces, in square units. A net, the solid unfolded flat, helps you list the faces. A prism's surface area is the two bases plus the lateral faces (perimeter × height). A cylinder's is 2πr² + 2πrh. A cone's is πr² + πrl, where l is the slant height, and a pyramid uses slant heights of its triangular faces. A sphere's surface area is 4πr².

Scaling solids follows the similarity rules. If every length is multiplied by k, surface area is multiplied by k² and volume by k³. Doubling the edges of a cube makes its surface area 4 times as large and its volume 8 times as large.

Density is mass ÷ volume, so mass = density × volume. Find the volume with the right formula first, then multiply, converting units if needed. Population density is people ÷ area, and comparing two regions means comparing their densities, not their raw totals. Rotating a rectangle about one of its sides forms a cylinder, rotating a right triangle about a leg forms a cone (that leg is the height, the other leg is the radius), and rotating a semicircle about its diameter forms a sphere. Identify the radius and height from the side lengths before applying the volume formula.

A cross-section is the shape you get when a plane slices a solid. A plane parallel to the base of a prism or cylinder gives a copy of the base, a plane perpendicular to the base gives a rectangle, and slices of a cone can give circles, ellipses or triangles. Each face the plane crosses adds one side, so a slice of a rectangular prism can have 3 to 6 sides, never 8. 1 liter = 1,000 cm³. Because volume is cubic, converting lengths by a factor k changes volume by k³: 1 ft³ = 12³ = 1,728 in³.

Key ideas

• Prism and cylinder: V = Bh (base area × height).
• Pyramid and cone: V = ⅓Bh.
• Sphere: V = (4/3)πr³, SA = 4πr².
• Cylinder surface area: 2πr² + 2πrh.
• Scaling lengths by k scales volume by k³.

Formulas and rules

Common mistakes

• Forgetting the ⅓ for cones and pyramids.
• Using diameter in place of radius.

Words to know

Volume of a prism or cylinder

Base area × height

Volume of a cone

⅓πr²h

Volume of a sphere

(4/3)πr³

Surface area of a sphere

4πr²

Scale factor k (volume)

Volumes scale by k³ (doubling lengths → 8×).

Prism

A solid with two congruent parallel polygon bases.

Pyramid volume

⅓Bh

Cylinder volume

πr²h

Cavalieri's principle

Solids with equal heights and equal cross-sections at every level have equal volumes.

Net

A flat pattern that folds into a solid.

Lateral area

The area of the sides, not including the bases.

Surface area of a cylinder

2πr² + 2πrh

Surface area of a cone

πr² + πrl (l = slant height)

Sample practice questions

37 more questions like these, from level 1 to 5, are in the practice set.