Find perimeter and area of plane figures

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Lesson progressPractice problems 0/3

Difficulty: Beginner
Estimated time: 31 minutes
Domains: Geometry and Trigonometry
Techniques: PerimeterAreaCircumferencePerpendicular-heightFormula-rearrangement

What you’ll learn

Tell perimeter, circumference, and area apart from the wording and the units.
Pick the right formula for rectangles, squares, triangles, parallelograms, trapezoids, whole circles, and the perimeter of a regular polygon.
Find the perpendicular height instead of grabbing a slanted side.
Put each number in its right place in the formula.
Run a formula backward to find a missing length.

Prerequisites

You don’t need any earlier lesson for this one. If you can do arithmetic with positive and negative numbers and plug numbers into a simple formula, you’re ready.

Why this matters on the SAT

Around or inside? Pick the measurement

The same two numbers can give very different answers. So before you calculate anything, ask what the question wants: the distance around the figure, or the space inside it?

Solution to the example

A border runs around the garden, so the question wants the perimeter:

\begin{aligned} P&=2\ell+2w\\[1.4em] &=2(12)+2(7)\\[1.4em] &=38. \end{aligned}

The answer is B. Choice D, $84$ , is $12\cdot7$ , the area. That counts the square feet of ground inside the garden, not the length of the border.

You can do this arithmetic in your head, so a calculator would only add steps. The real work was deciding what to calculate, and that’s the habit this whole topic runs on: decide first, then calculate.

SAT example

A border follows all four sides of the rectangle.

A rectangular garden is $12$ feet long and $7$ feet wide. A border will be placed around the garden. Which choice gives the length, in feet, of the border?

A
$19$
B
$38$
C
$70$
D
$84$

Choose perimeter or area first

Perimeter is the total distance around a polygon, so you add up every outside side. It’s a length, measured in linear units like feet or centimeters.

Circumference is the same idea for a circle: the distance around it. It’s a length too.

Area is the space inside a flat figure. It’s measured in square units, like square feet or $\text{cm}^2$ , because you’re counting how many small squares would cover it.

The wording usually tells you which one the question wants:

Words like around, border, fence, rim, and boundary point to perimeter or circumference.
Words like covers, inside, region, floor, and the surface of a flat figure point to area.

Units give you a second check. A length shouldn’t come out in square units, and an area shouldn’t come out in plain linear units.

Check your understanding:

A classroom needs trim around the edge of its floor and tile covering the floor. Which measurement does each job need, and what kind of units go with it?

Common mistake:

It’s tempting to add one length and one width, like $12+7=19$ for the garden. That covers only two of the four sides, so it’s half the perimeter. A rectangle has two lengths and two widths, so use $2\ell+2w$ or add all four sides.

Rectangles and squares

Perimeter runs along the outside edge. Area fills the shaded inside.

In the formulas below, $P$ stands for perimeter and $A$ stands for area.

A rectangle with length $\ell$ and width $w$ has two lengths and two widths around its edge, so

P=2\ell+2w.

Its area multiplies the two sides that meet at a right angle:

A=\ell w.

A square is a rectangle with four equal sides. If each side is $s$ , its perimeter and area are

P=4s \qquad\text{and}\qquad A=s^2.

These formulas also run backward. Say a rectangle has area $96$ square inches and length $12$ inches. Put what you know into $A=\ell w$ :

96=12w,

so $w=8$ inches. The area formula turned into an equation with one unknown side.

A square works the same way, with one twist. From $A=s^2$ , the side is a square root, and a real length can’t be negative, so take the positive root:

s=\sqrt{A}.

Check your understanding:

A square has area $121$ square centimeters. What are its side length and perimeter?

Perimeter of a regular polygon

Every side matches, so one labeled side tells you all six.

A regular polygon has all its sides the same length. So if it has $n$ sides, each of length $s$ , you don’t need to add them one at a time:

P=ns.

The hexagon above has $6$ sides of $7$ inches each, so

P=6(7)=42

inches.

This shortcut works only when every side matches. If the side lengths differ, add the actual outside sides instead.

Use a perpendicular height

The dashed height meets the base at a right angle. A slanted side isn’t the height.

For a triangle,

A=\frac12 bh.

For a parallelogram, a four-sided figure with two pairs of parallel sides,

A=bh.

The SAT reference sheet gives you the rectangle, triangle, and circle formulas, but not the area of a parallelogram or a trapezoid. Those two are yours to remember: $A=bh$ and $A=\frac12(b_1+b_2)h$ .

In both formulas, $b$ is the side you choose as the base, and $h$ is the perpendicular height, the distance to the base measured at a right angle. In a triangle, it runs from the corner opposite the base to the line the base sits on. In a parallelogram, it’s the shortest distance between the two parallel sides. On a figure, look for a right-angle marker or a dashed segment that meets the base squarely. In an obtuse triangle, one with an angle wider than $90^\circ$ , that segment can even land outside the triangle. A side is the height only when it meets the base at a right angle.

Why the $\frac12$ for a triangle? Two copies of the same triangle fit together into a parallelogram with the same base and height, so one triangle is half of it.

Common mistake:

A slanted side can look close enough to vertical to pass for the height. It isn’t. Area depends on how far the top is from the base, measured at $90^\circ$ to the base, not on how long the tilted side is. Find the segment that meets the base at a right angle.

Check your understanding:

A parallelogram has base $15$ centimeters, slanted side $9$ centimeters, and perpendicular height $7$ centimeters. Which two numbers go into $A=bh$ , and what is the area?

Example: Average the trapezoid’s bases

Worked example

The two parallel sides are the bases; the dashed perpendicular segment is the height.

A trapezoid’s parallel sides have lengths $10$ centimeters and $18$ centimeters. The perpendicular distance between them is $7$ centimeters. What is the area of the trapezoid, in square centimeters?

Step 1

Find the two bases

A trapezoid has one pair of parallel sides. Those are its bases, $b_1$ and $b_2$ . This is the formula the reference sheet leaves out:

A=\frac12(b_1+b_2)h.

Here, $b_1=10$ and $b_2=18$ .

Step 2

Use the perpendicular distance

The height is the perpendicular distance between the parallel bases, so $h=7$ . You don’t need the slanted sides at all.

Step 3

Substitute, then calculate

\begin{aligned} A&=\frac12(10+18)(7)\\[1.4em] &=\frac12(28)(7)\\[1.4em] &=14(7)\\[1.4em] &=98. \end{aligned}

The trapezoid’s area is $\boxed{98}$ square centimeters.

Look at the $\frac12(10+18)=14$ in there. That’s the average of the two bases, so here’s another way to think of the formula:

A=(\text{average base length})(\text{height}).

It’s the area of a rectangle $14$ wide and $7$ tall.

Step 4

Check the size

The average base, $14$ , sits between $10$ and $18$ . So the area should land between $10(7)=70$ and $18(7)=126$ , and $98$ does.

Check your understanding:

Say the same trapezoid kept both bases but had height $5$ instead of $7$ . What would its area be?

Whole circles

The radius goes from the center to the circle. The diameter goes all the way across, through the center.

The radius $r$ runs from the center to the circle. The diameter $d$ goes all the way across through the center, so it’s two radii long:

d=2r.

Circumference is the distance around the circle:

C=2\pi r=\pi d.

Area is the space inside it:

A=\pi r^2.

The area formula needs the radius, so if you’re given a diameter, halve it first. A circle with diameter $18$ centimeters has radius $9$ centimeters, so

A=\pi(9)^2=81\pi

square centimeters.

When the question or the answer choices use $\pi$ , leave $\pi$ in your answer. Swapping in $3.14$ turns an exact answer into an approximation the question didn’t ask for.

Check your understanding:

A circle has circumference $26\pi$ feet. What are its radius and area?

Common mistake:

Using the diameter as if it were the radius makes the area four times too big, because the doubled length gets squared. Before you plug in, check which segment you were given. If it goes all the way across through the center, divide by $2$ before squaring.

Choose the formula before the tool

Here’s the order that works for every one of these questions. The calculator doesn’t show up until step 5.

Name the figure, like “a parallelogram” or “a circle.”
Mark what the question wants: the perimeter, the circumference, the area, or a missing length.
Label each given number by its job, such as the base, the perpendicular height, the radius, or the diameter.
Pick the formula and put the numbers in.
Do the arithmetic the fastest reliable way, in your head for one familiar fact like $12\cdot7=84$ , or typed into Desmos as one complete expression when it isn’t that quick.
Finish with the right units, plain units for a length and square units for an area.

Desmos can’t do steps 1 to 4 for you. It can’t tell whether a slanted segment is the height, or whether the question wants the border or the inside. Once you’ve set things up, though, it saves you the messy arithmetic in between. Say you’ve found base $14.7$ and perpendicular height $8.6$ for $A=bh$ . Type 14.7*8.6 on one line, and Desmos gives $126.42$ . You multiplied two lengths in meters, so that’s square meters.

Calculator loads as you approach

You pick the formula and the height. Desmos does the multiplication.

Try it yourself:

Before you start each practice problem, write one word next to it: perimeter, circumference, area, or missing length. Make that call before any arithmetic.

Practice problems

Each of these uses a move you’ve just practiced. Keep an eye out for the slanted side in the second one.

Match the boundary and the inside

Practice problem

The cable follows the boundary; the tile covers the inside.

A rectangular patio is $15$ feet long and $8$ feet wide. A lighting cable will run once around its outside edge, and tile will cover its entire interior. Which choice gives the cable length first and the tile area second?

Calculator loads as you approach

Perimeter for the cable, area for the tile. These numbers are quick to do in your head.

Use a perpendicular height

Practice problem

Use the right-angle marker to identify the height.

The dimensions of a parallelogram are shown. Which choice gives its area, in square meters?

Calculator loads as you approach

Pick the height yourself, then type the whole product into Desmos.

Recover a trapezoid height

Practice problem

A trapezoid has area $96$ square inches. Its parallel bases have lengths $10$ inches and $14$ inches. What is the perpendicular height, in inches, of the trapezoid?

Your answer

Calculator loads as you approach

Put in the area and the bases, then solve for the height.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Perimeter and circumference measure the distance around a figure. Area measures the space inside.
Lengths come in linear units, and areas come in square units.
A rectangle has $P=2\ell+2w$ and $A=\ell w$ . A square has $P=4s$ and $A=s^2$ .
A triangle’s area is $\frac12 bh$ , and a parallelogram’s is $bh$ .
A trapezoid’s area is $\frac12(b_1+b_2)h$ , the average base length times the height.
For a triangle, parallelogram, or trapezoid, the height must meet the base at a right angle.
A regular polygon with $n$ equal sides of length $s$ has perimeter $ns$ .
For a circle, $d=2r$ , $C=2\pi r=\pi d$ , and $A=\pi r^2$ . Keep $\pi$ in the answer when the question or choices use it.
To find a missing length, put what you know into the formula and solve.
Decide what to calculate before you calculate. Do quick arithmetic in your head, type messier arithmetic into Desmos as one expression, and finish with the right units.

Next lesson

Break down composite and shaded figures

Combine or subtract familiar areas when one standard formula is not enough.

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Practice

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514 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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