Break down composite and shaded figures

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Lesson progressPractice problems 0/3

Difficulty: Intermediate
Estimated time: 27 minutes
Domains: Geometry and Trigonometry
Techniques: Composite-areaArea-additionArea-subtractionMissing-dimensions

What you’ll learn

Tell when one familiar formula won’t cover the region a question asks about.
Draw or spot the lines that split a figure into rectangles, triangles, and circle pieces.
Add pieces that don’t overlap, or subtract pieces that were cut out.
Find the lengths you need first, such as the inside of a frame.
Do longer arithmetic as one complete expression in Desmos.
Give lengths in units and areas in square units, keeping $\pi$ exact when the question needs it.

Prerequisites

Find perimeter and area of plane figures0%
Use standard perimeter and area formulas for rectangles, triangles, parallelograms, trapezoids, regular polygons, and circles.

Why this matters on the SAT

See the familiar shapes inside

The SAT often asks for the area of a region that isn’t one rectangle, triangle, or circle. The trick is to picture a complete familiar shape, then ask what was added to it or cut out of it.

SAT example

Complete the outer rectangle, then subtract the missing corner.

A rectangular floor is $14$ meters long and $9$ meters wide. A rectangular corner measuring $5$ meters by $4$ meters is removed, as shown. What is the area, in square meters, of the remaining floor?

A
$86$
B
$106$
C
$126$
D
$146$

Solution to the example

The L-shape has no formula of its own, but it’s easy to see as a whole rectangle with one corner cut out. So the floor is the whole minus the hole:

14(9)-5(4).

Type that whole expression into Desmos. It gives $106$ , with no subtotals to write down along the way.

\begin{aligned} A_{\text{remaining}} &=A_{\text{whole}}-A_{\text{corner}}\\[1.4em] &=106. \end{aligned}

The answer is B. Subtraction fits because the floor you want is what’s left once the corner is gone. Choice C, $126$ , is the whole rectangle before the corner comes out, and choice D, $146$ , adds the corner instead of taking it away.

Calculator loads as you approach

The picture says subtract. Desmos does the arithmetic in one line.

Choose add or subtract

Add pieces that build the region. Subtract pieces that were cut out of it.

A composite figure is made of two or more familiar shapes. Sometimes the line between the shapes is drawn for you, and sometimes you have to picture it.

When the region you want is built from pieces that don’t overlap, add them:

A_{\text{region}}=A_1+A_2+\cdots

Words like combined, joined, total, or made from often point to adding.

When the region you want is what’s left after something is taken out, subtract. That’s the whole minus the hole:

A_{\text{region}}=A_{\text{whole}}-A_{\text{removed}}.

Words like shaded, uncovered, outside, opening, border, remaining, or removed often point to subtracting. Treat those words as hints, not orders. Trace the region in the picture before you decide.

Check your understanding:

A rectangle and a triangle share one full edge without overlapping. Another figure is a circle with a smaller circle, sharing the same center, cut out of it. Which operation fits each figure?

Common mistake:

It’s tempting to add every area you can see, but that can count a region the question leaves out. Before you calculate, mark each region as in or out. Then add only the pieces that fill the region you want, or start from a complete whole and subtract what’s out.

Reveal familiar pieces

Here’s a short plan:

Mark the target. Lightly shade the region the question asks about, or trace its edge.
Pick a whole or draw a cut. Look for rectangles, triangles, trapezoids, circles, semicircles, or quarter-circles.
Find missing lengths first. A border on both sides changes a length twice.
Write one area expression. Get the whole setup down before any arithmetic.
Do the arithmetic the fastest way. Do a quick fact in your head, and type anything longer into Desmos as one line.
Check form and units. Lengths get plain units, and areas get square units. Keep $\pi$ in the answer unless the question asks for a decimal.

There’s often more than one good cut. The opening L-shape splits into two rectangles you can add, or it fills out to one big rectangle minus a corner. Pick the way with fewer pieces and fewer lengths to work out.

Try it yourself:

Go back to the opening L-shape and draw one vertical cut that splits it into two rectangles. Do their areas also add up to $106$ square meters?

Example: Find the area of a uniform frame

Worked example

The frame takes

2

feet from both ends of each inner dimension.

A rectangular frame has outer dimensions of $18$ feet by $12$ feet. The frame is uniformly $2$ feet wide on all four sides. What is the area of the frame material, in square feet?

Step 1

Name the whole and the opening

The frame is everything inside the outer rectangle but outside the opening. That’s the whole minus the hole:

A_{\text{frame}}=A_{\text{outer}}-A_{\text{opening}}.

Step 2

Find the opening’s size

The $2$ -foot frame sits on the left and on the right, so the opening’s length loses $2$ feet twice:

18-2-2=14\text{ feet}.

The frame also sits on the top and on the bottom, so the opening’s width loses $2$ feet twice too:

12-2-2=8\text{ feet}.

The opening is $14$ feet by $8$ feet.

Step 3

Subtract the two rectangle areas

With the lengths settled, type the whole expression 18(12)-14(8) into Desmos.

Calculator loads as you approach

The outer area minus the opening, all in one expression. Reset brings back this calculation.

\begin{aligned} A_{\text{frame}} &=18(12)-14(8)\\[1.4em] &=104. \end{aligned}

The area of the frame material is $\boxed{104}$ square feet.

Step 4

Check the region and units

The whole outer rectangle is $216$ square feet, and the frame covers only part of it, so an answer between $0$ and $216$ makes sense. The question asks for an area, so the unit is square feet.

Check your understanding:

What if the question asked for the area of the opening instead?

Common mistake:

If you take the frame width off each outer length only once, you get an opening of $16$ by $10$ and a frame of only $56$ square feet. But the frame sits at both ends of each length. Sketch the opening and label each direction $2+\text{opening}+2$ before you calculate.

Keep circle pieces exact

A ring is the outer circle with the inner circle taken out.

Circle pieces use the same add-or-subtract choice. A ring is whole minus hole. For a ring with outer radius $R$ and inner radius $r$ ,

\begin{aligned} A_{\text{ring}} &=\pi R^2-\pi r^2\\[1.4em] &=\pi(R^2-r^2). \end{aligned}

When a path of width $w$ goes around a circle of radius $r$ , the outer radius is $r+w$ . The width gets added only once, along a radius. That’s different from a rectangular frame, where the width shows up at both ends of a side.

A semicircle is half a circle, so its area is $\frac12\pi r^2$ . A quarter-circle’s area is $\frac14\pi r^2$ . For composite areas, those fractions are all you need. Arc length, sectors from an angle, and radians come later, in their own lesson.

Check your understanding:

A circular pond has radius $6$ meters, and a path $3$ meters wide goes around it. Write an expression for the path’s area, and leave it in terms of $\pi$ .

Common mistake:

Subtracting the radii before squaring gives the area of one circle whose radius is the difference, not the area between two circles. With $R=5$ and $r=3$ , $\pi(5-3)^2$ is only $4\pi$ , but the ring is $25\pi-9\pi=16\pi$ . Square each radius first: $\pi R^2-\pi r^2$ .

Choose the geometry, then the calculation

In both worked examples, the work split the same way: you figure out the shape, and Desmos does the arithmetic.

The calculator can’t tell that a border comes off both ends of a side. It can’t see which corner triangles sit outside a polygon on a grid, and it can’t decide whether a shaded region means add or subtract. Those choices come from the picture, so they’re yours. Once the setup is right, one expression like 18(12)-14(8) is faster and safer than working out each product and then combining them.

Two kinds of work are better kept off the calculator:

If the arithmetic is quick in your head, like $60+20$ , do it there.
If the answer has $\pi$ in it, keep $\pi$ exact. Two quarter-circles of radius $7$ have area exactly $\frac{49\pi}{2}$ , and a decimal like $76.97$ won’t match answer choices written with $\pi$ .

Practice problems

These go from adding two pieces, to subtracting curved pieces, to boxing in a shape on a coordinate grid. Before each one, decide whether its arithmetic is quick enough to do by hand.

Add two regions

Practice problem

The rectangle and triangle share an edge but do not overlap.

The front of a building is formed by a rectangle and a triangular roof, as shown. The regions do not overlap. What is the total area, in square feet, of the building front?

Calculator loads as you approach

Write both areas first. This arithmetic is quicker by hand.

Subtract curved regions

Practice problem

Together, two quarter-circles have the area of half a circle with radius

7

Rectangle $ABCD$ is $14$ units long and $10$ units wide. Two nonoverlapping quarter-circles of radius $7$ are removed from opposite corners, as shown. The remaining region $R$ is shaded. Which choice gives the area, in square units, of region $R$ ?

Calculator loads as you approach

The choices keep pi, so you should too.

Enclose a coordinate polygon

Practice problem

Enclose the octagon in a rectangle, then account for the four missing corners.

The shaded octagon has vertices

(-5,1),\ (-2,4),\ (4,4),\ (7,1),\ (7,-3),\ (4,-6),\ (-2,-6),\ (-5,-3)

in order around its boundary. What is its area, in square units?

Your answer

Calculator loads as you approach

Read the lengths from the coordinates, then enter the whole rectangle-minus-triangles expression.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

A composite figure hides familiar shapes. Find them first.
Add pieces that build the region without overlapping.
Subtract pieces cut out of a complete whole: the whole minus the hole.
Find missing lengths before area. A uniform border on a rectangle comes off both ends of each length.
A semicircle is $\frac12\pi r^2$ and a quarter-circle is $\frac14\pi r^2$ .
For a polygon on a grid, you can box it in a rectangle and subtract the corner triangles.
You choose the pieces and lengths, and Desmos does longer arithmetic as one expression.
Do quick facts in your head instead.
Answer with what the question asks for, in the right units, and keep $\pi$ exact unless it asks for a decimal.

Next lesson

Find surface area and volume

Extend familiar area formulas to the surfaces and volumes of three-dimensional solids.

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Practice

Practice this lesson

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

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