Unit 5, lesson 55

Area, Perimeter and Composite Figures

How to find the area of an odd shape by splitting it into simple pieces or subtracting a piece, why a triangle needs its perpendicular height, and how perimeter differs from area.

  • Theory part 109
  • Practice part 110
  • Area, Volume and Scaling

Theory, part 109

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Practice, part 110

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Painting a wall and putting trim around it are two different jobs. The paint covers the surface, so you need its area. The trim follows the edge, so you need its perimeter. Real walls have windows and doors, which turns a rectangle into a composite figure, and the work becomes splitting the shape into pieces you know or cutting out the parts that do not count.

See also Geometry Formulas and Units in Foundations for Calculus.

Key terms

Area

Definition The area of a plane region is the number of unit squares needed to cover it, with fractional squares allowed. Area has two basic properties. Congruent regions have equal area, and if a region is split into pieces that do not overlap, its area is the sum of the areas of the pieces. A rectangle with length ℓ\ell and width ww has area A=ℓwA = \ell w.

In plain words Area measures how much surface a shape covers. Cut a shape into pieces without overlapping and the pieces add back up to the whole.

Example A rectangle has area 8484 square feet and length 1212 feet. From 12w=8412w = 84 the missing width is w=7w = 7 feet. Its perimeter is 2(12+7)=382(12 + 7) = 38 feet, a completely different number with different units.

Perimeter

Definition The perimeter of a polygon is the sum of the lengths of its sides. More generally, the perimeter of a plane region is the length of its boundary. For a circle the perimeter is the circumference 2πr2\pi r.

In plain words Perimeter is the distance you walk if you trace the edge once all the way around. It is a length, not a surface.

Example A 1010 by 88 rectangle with a 44 by 33 rectangle cut out of one corner forms an L shape with sides 1010, 55, 44, 33, 66 and 88. Its perimeter is

10+5+4+3+6+8=36,10 + 5 + 4 + 3 + 6 + 8 = 36,

the same as the full rectangle, because the two new edges of the notch exactly replace the edges that were removed. The area did change, as the composite figure section shows.

Perpendicular height

Definition For a chosen base of a triangle, the height is the length of the segment from the opposite vertex to the line containing the base, perpendicular to that line. With base bb and that height hh,

A=12bh.A = \frac{1}{2} b h.

A parallelogram with base bb and perpendicular distance hh between that base and the opposite side has area A=bhA = bh. A diagonal splits it into two congruent triangles, each with base bb and height hh, so the area is 2⋅12bh=bh2 \cdot \frac{1}{2} bh = bh. You may also have seen a triangle cut off one end and slid to the other to make a bb by hh rectangle. That picture works when the perpendicular from a top corner lands on the base, and a very slanted parallelogram needs more cuts, so the diagonal argument is the safer one. The SAT reference sheet lists the rectangle and triangle formulas but not the parallelogram or trapezoid formulas. A slanted side is not a height unless it meets the base at a right angle. The height may fall outside the triangle when the triangle has an obtuse angle.

In plain words Height always means straight up from the base at a right angle, like dropping a plumb line. The slanted edge is longer than the height, so using it makes the area too big.

Example An isosceles triangle has sides 1313, 1313 and 1010. The height to the base of 1010 lands at the midpoint of the base and makes a right triangle with legs 55 and hh and hypotenuse 1313, so

h=132−52=12.h = \sqrt{13^2 - 5^2} = 12.

The area is 12⋅10⋅12=60\frac{1}{2} \cdot 10 \cdot 12 = 60. Using the slanted side 1313 as the height would give 6565, which is wrong.

Composite figure

Definition A composite figure is a region made of simpler shapes such as rectangles, triangles and parts of circles. If it is split into pieces that do not overlap and together fill it, its area is the sum of the pieces. Formulas that are not on the SAT reference sheet can be found this way. For example, a trapezoid with parallel bases b1b_1 and b2b_2 and height hh splits along a diagonal into two triangles, so

A=12b1h+12b2h=12(b1+b2)h.A = \frac{1}{2} b_1 h + \frac{1}{2} b_2 h = \frac{1}{2}(b_1 + b_2) h.

In plain words Break the shape into pieces you already know, find each area, and add. Choose cuts that make the missing lengths easy to read.

Example The L shape from the perimeter section can be split by a horizontal cut into a 1010 by 55 rectangle and a 66 by 33 rectangle, so its area is 50+18=6850 + 18 = 68. A trapezoid with bases 66 and 1414 and height 55 has area 12(6+14)⋅5=50\frac{1}{2}(6 + 14) \cdot 5 = 50.

Subtraction of areas

Definition If region SS lies entirely inside region RR, the area of the part of RR outside SS is

area(R)−area(S).\text{area}(R) - \text{area}(S).

Subtract only a region that is really inside the larger one and is meant to be removed, and never subtract the same part twice.

In plain words Sometimes it is easier to take the big simple shape and cut out what does not belong, like a wall minus its window.

Example A circle of radius 55 fits exactly inside a square with side 1010. The region inside the square but outside the circle has area

100−25π≈21.46.100 - 25\pi \approx 21.46.

The L shape works this way too. The full rectangle minus the notch is 80−12=6880 - 12 = 68, matching the sum from before.

Units

Definition Lengths and perimeters are measured in linear units such as meters. Areas are measured in square units such as square meters, the area of a square whose side is one unit. When a length scale changes by a factor, the area scale changes by the square of that factor, so 11 meter =100= 100 centimeters gives

1 m2=1002 cm2=10,000 cm2.1 \text{ m}^2 = 100^2 \text{ cm}^2 = 10{,}000 \text{ cm}^2.

In plain words Perimeter answers end in units and area answers end in square units. Converting area units means converting both dimensions.

Example A floor is 44 meters by 33 meters. Its perimeter is 1414 meters and its area is 1212 square meters. In centimeters the floor is 400400 by 300300, so the area is 120,000120{,}000 square centimeters, not 1,2001{,}200.

Check yourself

  1. A rectangle has an area of 9696 square centimeters and a width of 66 centimeters. What is the perimeter of the rectangle, in centimeters? (A) 2222, (B) 3232, (C) 4444, (D) 9696
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The answer is (C). The length is 96÷6=1696 \div 6 = 16, so the perimeter is 2(16+6)=442(16 + 6) = 44 centimeters. Choice (A) adds only one length and one width, choice (B) doubles the length alone, and choice (D) is the area.

  1. A rectangle is 1212 inches long and 99 inches wide. A right triangle with legs of 44 inches and 66 inches is cut from one corner, with its legs along two sides of the rectangle. What is the area, in square inches, of the remaining figure?
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The answer is 9696. The rectangle has area 12⋅9=10812 \cdot 9 = 108 and the triangle has area 12⋅4⋅6=12\frac{1}{2} \cdot 4 \cdot 6 = 12, so the remaining area is 108−12=96108 - 12 = 96. Forgetting the 12\frac{1}{2} would give 8484.

  1. An isosceles triangle has two sides of length 1010 and a base of length 1212. What is the area of the triangle? (A) 4848, (B) 6060, (C) 9696, (D) 120120
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The answer is (A). The height to the base meets it at its midpoint, so it is a leg of a right triangle with hypotenuse 1010 and other leg 66, which gives h=100−36=8h = \sqrt{100 - 36} = 8. The area is 12⋅12⋅8=48\frac{1}{2} \cdot 12 \cdot 8 = 48. Choice (B) uses the slanted side 1010 as the height, choice (C) forgets the 12\frac{1}{2}, and choice (D) does both.

How this lesson connects