Topic 7 of 7

Area and volume

Step 12 · Geometry topics

Find the areas of triangles, parallelograms, trapezoids and regular polygons, the volumes of prisms, cylinders, pyramids, cones and spheres, and how area and volume change when a shape is scaled.

Areas of polygons

Many area formulas come from the rectangle: cut a shape apart and rearrange it into a rectangle, or see it as half of one.

Shape	Area
parallelogram	base × height
triangle	$\frac{1}{2} \times \text{base} \times \text{height}$
trapezoid	$\frac{1}{2}(b_1 + b_2) \times h$
regular polygon	$\frac{1}{2} \times \text{apothem} \times \text{perimeter}$

A parallelogram: cut off the triangle at one end along the dashed height and slide it to the other end to make a rectangle with the same base and height.

Example: A trapezoid

A trapezoid has parallel sides of 8 and 14 and a height of 5. What is its area?

The average of the parallel sides is 11.
11 × 5 = 55.

Answer55

Example: A regular hexagon

A regular hexagon has sides of 6 and an apothem (center to the middle of a side) of about 5.2. What is its area?

The perimeter is 6 × 6 = 36.
Half of the apothem times the perimeter: half of 187.2, which is 93.6.

AnswerAbout 93.6

Volumes of prisms, cylinders, pyramids and cones

Prisms and cylinders have the same cross-section all the way up, so volume = base area × height. Pyramids and cones come to a point and hold exactly a third as much.

Solid	Volume
prism	Bh
cylinder	πr²h
pyramid	$\frac{1}{3}Bh$
cone	$\frac{1}{3}\pi r^{2}h$

B is the area of the base and h is the height, measured straight up.

Example: A cylinder

A can has a radius of 4 cm and a height of 10 cm. What is its volume? Use 3.14.

πr²h is π × 16 × 10, which is 160π.
160 × 3.14 = 502.4.

Answer502.4 cm³

Example: A cone

A cone has the same radius and height as the can. What is its volume?

A third of the cylinder: 160π ÷ 3.
That is about 167.5.

AnswerAbout 167.5 cm³

A 3 by 3 by 4 prism holds 36 cubes. A pyramid on the same base with the same height holds a third as much: 12.

Spheres and scale factors

A sphere of radius r has these volume and surface area formulas:

V = \frac{4}{3}\pi r^{3} \qquad A = 4\pi r^{2}

Example: A sphere

A ball has a radius of 3 in. What is its volume? Give it in terms of π.

$\frac{4}{3}\pi \times 27 = 36\pi$ .

Answer36π in³, about 113 in³

When a solid is scaled by a factor k, lengths are multiplied by k, areas by k², and volumes by k³.

Scale factor	Lengths	Areas	Volumes
2	× 2	× 4	× 8
3	× 3	× 9	× 27
$\frac{1}{2}$	$\times \frac{1}{2}$	$\times \frac{1}{4}$	$\times \frac{1}{8}$

Example: Scaling a model

A model statue is built at a scale of 1 : 10. The model uses 2 liters of clay. How much would the full-size statue use?

Volumes scale by 10³ = 1,000.
2 × 1,000 = 2,000.

Answer2,000 liters

Try it yourself

Work each one on paper first, then open the answer to check.

A triangle has base 12 and height 7. What is its area?
Show answer
42
A trapezoid has parallel sides of 6 and 10 and a height of 4. What is its area?
Show answer
32
A box is 5 by 4 by 3. What is its volume?
Show answer
60
A cylinder has radius 3 and height 7. What is its volume, in terms of π?
Show answer
63π
A square pyramid has a 6 by 6 base and a height of 10. What is its volume?
Show answer
120
A sphere has radius 6. What is its volume, in terms of π?
Show answer
288π
Two similar solids have scale factor 4. How many times the volume is the larger?
Show answer
64