SSAT Upper Level Math : Areas and Perimeters of Polygons

Study concepts, example questions & explanations for SSAT Upper Level Math

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Example Questions

Example Question #101 : Areas And Perimeters Of Polygons

Parallelogram

Figure NOT drawn to scale.

The above depicts Rhombus \displaystyle ABCD, which has perimeter 80. \displaystyle XC = 16.

Give the area of  Rhombus \displaystyle ABCD.

Possible Answers:

\displaystyle 320

\displaystyle 64

\displaystyle 256

\displaystyle 400

Correct answer:

\displaystyle 320

Explanation:

The area of any parallelogram is the product of the length of its base and that of a corresponding altitude. We can take \displaystyle \overline{CD} as a base and perpendicular \displaystyle \overline{XC} as an altitude.

All four sides of a rhombus have the same length, so we can find \displaystyle CD by dividing the perimeter - the sum of the lengths of the four sides - by 4:

\displaystyle CD = p \div 4 = 80 \div 4 = 20

Now multiply the lengths of this base and the altitude to get the area:

\displaystyle a = CD \cdot XC = 20 \cdot 16 = 320

Example Question #1 : Understand Categories And Subcategories Of Two Dimensional Figures: Ccss.Math.Content.5.G.B.3

What two shapes can a square be classified as? 

Possible Answers:

Trapezoid and Rhombus 

Rectangle and Rhombus 

Rhombus and Triangle 

Trapezoid and Triangle 

Rectangle and Triangle 

Correct answer:

Rectangle and Rhombus 

Explanation:

A square can also be a rectangle and a rhombus because a rectangle has to have at least \displaystyle 2 sets of equal side lengths and a rhombus has to have \displaystyle 4 equal side lengths, like a square, and at least \displaystyle 2 sets of equal angles.

Example Question #1 : How To Find The Perimeter Of A Parallelogram

The base length of a parallelogram is \displaystyle 4t+6 which is two times more than its side length. Give the perimeter of the parallelogram in terms of \displaystyle t.

Possible Answers:

\displaystyle 12t+9

\displaystyle 12t+18

\displaystyle 10t+18

\displaystyle 8t+9

\displaystyle 6t+9

Correct answer:

\displaystyle 12t+18

Explanation:

The side length is half of the base length:

\displaystyle h=\frac{w}{2}=\frac{4t+6}{2}=\frac{4t}{2}+\frac{6}{2}=2t+3

The perimeter of a parallelogram is:

\displaystyle Perimeter=2(w+h)

Where:


  \displaystyle w is the base length of the parallelogram and \displaystyle h is the side length

 

\displaystyle =2[(4t+6)+(2t+3)]

\displaystyle =2(6t+9)

\displaystyle =12t+18

Example Question #2 : How To Find The Perimeter Of A Parallelogram

The side length of a parallelogram is \displaystyle t^2+1 and the base length is three times more than side length. Give the perimeter of the parallelogram in terms of \displaystyle t.

Possible Answers:

\displaystyle 8t^2+8

\displaystyle 8t^2

\displaystyle 4t^2+8

\displaystyle 4t^2+4

\displaystyle 8t^2+4

Correct answer:

\displaystyle 8t^2+8

Explanation:

The base length is three times more than the side length, so we have:

 

Base length \displaystyle =3(t^2+1)=3t^2+3

 

The perimeter of a parallelogram is:

\displaystyle 2(w+h)

Where:

\displaystyle w is the base length of the parallelogram and \displaystyle h is the side length. So we get:

\displaystyle Perimeter=2(w+h)=2\left [ (3t^2+3)+(t^2+1) \right ]

\displaystyle =2(4t^2+4)

\displaystyle =8t^2+8

Example Question #1 : How To Find The Perimeter Of A Parallelogram

The base length of a parallelogram is 10 inches and the side length is 6 inches. Give the perimeter of the parallelogram.

Possible Answers:

\displaystyle 34

\displaystyle 16

\displaystyle 32

\displaystyle 30

\displaystyle 60

Correct answer:

\displaystyle 32

Explanation:

Like any polygon, the perimeter of a parallelogram is the total distance around the outside, which can be found by adding together the length of each side. In case of a parallelogram, each pair of opposite sides is the same length, so the perimeter is twice the base plus twice the side length. Or as a formula we can write:

 

\displaystyle Perimeter=2(w+h)

Where:

\displaystyle w is the base length of the parallelogram and \displaystyle h is the side length. So we can write:

 

\displaystyle Perimeter=2(w+h)=2(10+6)=32

 

Example Question #1 : How To Find The Perimeter Of A Parallelogram

The base length of a parallelogram is \displaystyle t+2. If the perimeter of the parallelogram is 24, give the side length in terms of \displaystyle t.

Possible Answers:

\displaystyle 20+t

\displaystyle 10+t

\displaystyle 10-t

\displaystyle 22-2t

\displaystyle 20-t

Correct answer:

\displaystyle 10-t

Explanation:

Let:

Side length \displaystyle =x.

The perimeter of a parallelogram is:

\displaystyle 2w+2h

where:

\displaystyle w is the base length of the parallelogram and \displaystyle h is the side length. The perimeter is known, so we can write:

 

\displaystyle Perimeter=24=2(t+2)+2x

 

Now we solve the equation for \displaystyle x:

 

\displaystyle 24=2(t+2)+2x

\displaystyle \Rightarrow 24=2t+4+2x

\displaystyle \Rightarrow 24-2t-4=2x

\displaystyle \Rightarrow 20-2t=2x

\displaystyle \Rightarrow 10-t=x

 

Example Question #2 : How To Find The Perimeter Of A Parallelogram

The base length of a parallelogram is identical to its side length. If the perimeter of the parallelogram is 40, give the base length.

Possible Answers:

\displaystyle 10

\displaystyle 20

\displaystyle 16

\displaystyle 8

\displaystyle 5

Correct answer:

\displaystyle 10

Explanation:

The perimeter of a parallelogram is:

\displaystyle 2(w+h)

Where:

\displaystyle w is the base length of the parallelogram and \displaystyle h is the side length. In this problem the base length and side length are identical, that means:

\displaystyle w=h

So we can write:

\displaystyle Perimeter=40=2(w+w)\Rightarrow 20=2w\Rightarrow w=10

Example Question #841 : Geometry

The base length of a parallelogram is \displaystyle 2t+3 and the side length is \displaystyle 2t-3. Give the perimeter of the parallelogram in terms of \displaystyle t and calculate it for \displaystyle t=1.

Possible Answers:

\displaystyle 8t, 8

\displaystyle 6t+6, 12

\displaystyle 6t, 6

\displaystyle 8t+8, 16

\displaystyle 4t, 4

Correct answer:

\displaystyle 8t, 8

Explanation:

The perimeter of a parallelogram is:

\displaystyle 2(w+h)

where:

\displaystyle w is the base length of the parallelogram and \displaystyle h is the side length. So we have:

 

\displaystyle Perimeter=2(w+h)=2\left [ (2t+3)+(2t-3) \right ]=2(4t)=8t

and:

\displaystyle t=1\Rightarrow 8t=8\times 1=8

Example Question #1061 : Ssat Upper Level Quantitative (Math)

Parallelogram1

The above parallelogram has area 100. Give its perimeter.

Possible Answers:

\displaystyle 20+20\sqrt{2}

\displaystyle 20+20\sqrt{3}

\displaystyle 40\sqrt{2}

\displaystyle 40\sqrt{3}

\displaystyle 40

Correct answer:

\displaystyle 20+20\sqrt{2}

Explanation:

The height of the parallelogram is \displaystyle BD, and the base is \displaystyle DC. By the \displaystyle 45^{\circ}-45^{\circ}-90^{\circ} Theorem, \displaystyle BD=CD. Since the product of the height and the base of a parallelogram is its area, 

\displaystyle BD \cdot CD = A

\displaystyle \left (BD \right )^{2} = 100

\displaystyle BD = 10

By the \displaystyle 45^{\circ}-45^{\circ}-90^{\circ} Theorem, 

\displaystyle CD = AB = BD = 10, and

\displaystyle AD = BC = BD \cdot \sqrt{2} = 10\sqrt{2}

The perimeter of the parallelogram is

\displaystyle AB + CD + BC + AD = 10 + 10 + 10\sqrt{2}+ 10\sqrt{2} = 20+20\sqrt{2}

Example Question #842 : Geometry

Parallelogram2

Give the perimeter of the above parallelogram if \displaystyle BD = 10.

Possible Answers:

\displaystyle 40+20\sqrt{3}

\displaystyle 60\sqrt{2}

\displaystyle 40+40\sqrt{3}

\displaystyle 60\sqrt{3}

\displaystyle 40+20\sqrt{2}

Correct answer:

\displaystyle 40+20\sqrt{3}

Explanation:

By the \displaystyle 30^{\circ}-60^{\circ}-90^{\circ} Theorem:

\displaystyle AB = CD= BD\sqrt{3}= 10\sqrt{3}, and

\displaystyle AD=BC= 2\cdot BD = 2\cdot 10 = 20

The perimeter of the parallelogram is

\displaystyle AB + CD + BC + AD = 20+20+ 10\sqrt{3}+ 10\sqrt{3} = 40+20\sqrt{3}

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