Chapter 09 of 10

Geometry & Measurement

How can complex shapes be measured by breaking them into familiar pieces?

Measurement formulas describe how dimensions combine. Area covers a flat region, volume fills a solid, and surface area covers every outside face. Composite figures become manageable when they are decomposed into simpler shapes.

01

The big ideas

Understand these before you worry about speed.

Area uses square units

A triangle is half of a related parallelogram, and other polygons can be split into rectangles and triangles. Use perpendicular height, not a slanted side.

ExampleA triangle with b=10\displaystyle b=10 and h=7\displaystyle h=7 has A=12bh=35\displaystyle A=\frac{1}{2}bh=35 square units.

Volume counts cubic units

A rectangular prism's volume is length times width times height. Fractional edge lengths follow the same formula.

ExampleV=lwh=4(3)(2.5)=30\displaystyle V=lwh=4(3)(2.5)=30 cubic units.

Surface area covers every exposed face

Use a net or list the faces to avoid missing or double-counting one. Congruent opposite faces have equal areas.

ExampleA 2×3×4\displaystyle 2\times3\times4 prism has SA=2(6)+2(8)+2(12)=52\displaystyle SA=2(6)+2(8)+2(12)=52 square units.

Formulas can reveal missing dimensions

Substitute every known measurement, then solve the resulting equation for the unknown dimension. Include units in the final answer.

ExampleIf V=72\displaystyle V=72, l=6\displaystyle l=6, and w=3\displaystyle w=3, then 72=18h\displaystyle 72=18h, so h=4\displaystyle h=4.
Visual models

See what each measurement counts

Labels show the perpendicular dimensions used by each formula. The drawings are not to scale.

Triangle areaThe perpendicular height forms half of a related parallelogram.A = ½bh = 35 square units
Rectangular-prism volumeMultiply three perpendicular dimensions to count the cubic units inside.V = lwh = 30 cubic units
Surface area from a netA closed prism has three pairs of congruent opposite faces.SA = 2lw + 2lh + 2wh
Composite areaDraw a dividing line, find each familiar area, and combine the pieces.A = A₁ + A₂
02

Worked example

Follow the reason for each step—not just the symbols.

Problem

A rectangular prism is 512\displaystyle 5\frac{1}{2} feet long, 3\displaystyle 3 feet wide, and 2\displaystyle 2 feet high. Find its volume.

  1. Use V=lwh\displaystyle V=lwh.
  2. Substitute: V=512(3)(2)\displaystyle V=5\frac{1}{2}(3)(2).
  3. Multiply efficiently: 3(2)=6\displaystyle 3(2)=6, then 512(6)=33\displaystyle 5\frac{1}{2}(6)=33.
  4. Label the result with cubic feet because three dimensions were multiplied.
Answer33 ft3\displaystyle 33\text{ ft}^3
03

Try it yourself

Work on paper first. Open each answer only when you are ready to check.

1

Find the area of a triangle with base 14\displaystyle 14 cm and height 9\displaystyle 9 cm.

Check answer63 cm2\displaystyle 63\text{ cm}^2
2

Find the volume of a 4×6×2.5\displaystyle 4\times6\times2.5 rectangular prism.

Check answer60\displaystyle 60 cubic units
3

A rectangle has area 54 m2\displaystyle 54\text{ m}^2 and width 6 m\displaystyle 6\text{ m}. Find its length.

Check answer9 m\displaystyle 9\text{ m}
4

Find the surface area of a cube with side length 5\displaystyle 5.

Check answer150\displaystyle 150 square units
04

Practice and tools

Use the resource that matches what you need next.

Worksheet

Area Practice

Find areas of familiar and composite figures.

Open resource Worksheet

Volume with Cubes

Connect cubic units to rectangular-prism volume.

Open resource Worksheet

Surface Area

Measure cubes, prisms, and open figures.

Open resource Worksheet

Measurement Conversions

Convert measurements while preserving meaning.

Open resource