Mathematics
Squares
In mathematics, a square is a quadrilateral with four equal sides and four right angles. The area of a square is calculated by squaring the length of one of its sides. The perimeter of a square is found by multiplying the length of one side by 4.
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No longer available |Learn more- Daniel C. Alexander, Geralyn M. Koeberlein, , , Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
All right are 7. SAS 8. CPCTC ∠ s ∠ s ∠ s A square is a rectangle that has two congruent adjacent sides. (See Figure 4.22.) DEFINITION COROLLARY 4.3.3 All sides of a square are congruent. Because a square is a type of rectangle, it has four right angles and its diagonals are con-gruent. Because a square is also a parallelogram, its opposite sides are parallel. For any square, we can show that the diagonals are perpendicular; see Exercise 32. In Chapter 8, we measure area in “square units.” For the calculation of area, we count the number of congruent Squares that fit inside a geometric region. THE RHOMBUS The next type of quadrilateral we consider is the rhombus. The plural of the word rhombus is rhombi (pronounced rho ˘ m-bi ¯ ). D iscover Given a rectangle MNPQ (like a sheet of paper), draw diagonals and . From a second sheet, cut out (formed by two sides and a diagonal of MNPQ ). Can you position so that it coincides with ? ANSWER NQP MPQ MPQ NQ MP Yes A B D C Square ABCD EXS. 1–4 Figure 4.22 EXS. 5–7 80'' 72'' A carpenter has installed the frame for a patio door. To check the frame for “square-ness,” he measures the lengths of the two diagonals of the rectangular opening…to be sure that the lengths are equal and that the opening is actually a rectangle. Geometry in the Real World A rhombus is a parallelogram with two congruent adjacent sides. (See Figure 4.23.) DEFINITION Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. In Figure 4.23, the adjacent sides and of rhombus ABCD are marked congru-ent.
eBook - PDF- (Author)
- 2015(Publication Date)
- For Dummies(Publisher)
In the previous section, the perimeter formulas deal with linear measure. Linear measure is just one dimension. It’s from one place to another — there’s no breadth to it. You measure it with a ruler or tape measure in one direction. Square measurements are used to measure area. Area takes two measures — one along a side and a second perpendicular (90 degrees) to that side. Laying out rectangles and Squares Rectangles and Squares have basically the same area formulas because they both have square corners and the equal lengths on opposite sides. The general procedure here is just to multiply the measure of the length times the measure of the width. The product of two sides that are next to one another is the area. Finding the area of a rectangle or square Most rooms in homes, schools and offices are rectangular in shape. Desks and tables and rugs are usually rectangular, also. This makes it easy to fit furniture and other objects in the room. The area of a rectangle is its length times its width, and the area of a square is the square of the measure of any side: Rectangle: A = l × w Square: A = s 2 Say a garden 35 metres long by 15 metres wide needs some fertiliser. If a bag of fertiliser covers 6 square metres, how much fertiliser do you need? First determine how many square metres the garden is. area of garden = l × w = 35 × 15 = 525 square metres Divide the 525 square metres by 6 square metres: 525 ÷ 6 = 87.5 square metres You can buy 88 bags and have some left over, or buy 87 bags and skimp a little in some places. 322 Part IV: Applying Algebra and Understanding Geometry Tuning in triangles Finding the area of a triangle can be a bit of a challenge. Basically, a triangle’s area is half that of an imaginary rectangle that the triangle fits into. However, it isn’t always easy or necessary to find the length and width of this hypothetical rectangle — you just need a measurement or two from the triangle.
eBook - PDF- Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
uni2220NMQ _ uni2220PQM 7. uni25B3NMQ _ uni25B3PQM 8. MP _ QN 1. Given 2. By definition, a rectangle is a uni25B1 with a right angle 3. Opposite sides of a uni25B1 are _ 4. Identity 5. By Corollary 4.3.1, the four uni2220s of a rectangle are right uni2220s 6. All right uni2220s are _ 7. SAS 8. CPCTC A B D C Square ABCD Figure 4.22 200 CHAPTER 4 ■ QUADRILATERALS Because a square is a type of rectangle, it has four right angles and its diagonals are congruent. Because a square is also a parallelogram, its opposite sides are parallel. For any square, we can show that the diagonals are perpendicular; see Exercise 34. In Chapter 8, we measure area in “square units.” For the calculation of area, we count the number of congruent Squares (square units) that fit inside a geometric region. THE RHOMBUS The next type of quadrilateral we consider is the rhombus. The plural of the word rhombus is rhombi (pronounced rho ˘m-bi ¯ ). While this definition requires only two congruent sides, the following corollary deter- mines that a square actually has four congruent sides. Corollary 4.3.3 All sides of a square are congruent. Corollary 4.3.4 All sides of a rhombus are congruent. Corollary 4.3.5 The diagonals of a rhombus are perpendicular. SSG EXS. 5–7 A rhombus is a parallelogram with two congruent adjacent sides. (See Figure 4.23.) DEFINITION In Figure 4.23, the adjacent sides AB and AD of rhombus ABCD are marked congru- ent. Because a rhombus is a type of parallelogram, it is also necessary that AB _ DC and AD _ BC . Thus, we have Corollary 4.3.4. We will use Corollary 4.3.4 in the proof of the following corollary. To visualize Corollary 4.3.5, see Figure 4.24(a). Discover What type of quadrilateral is deter- mined when an isosceles triangle is reflected across its base? ANSWER Rhombus EXAMPLE 2 Study the picture proof of Corollary 4.3.5. In the proof, pairs of triangles are congruent by the reason SSS.
eBook - ePubMastering Math Manipulatives, Grades K-3
Hands-On and Virtual Activities for Building and Connecting Mathematical Ideas
- Sara Delano Moore, Kimberly Rimbey(Authors)
- 2021(Publication Date)
- Corwin(Publisher)
In this activity, students use the Make a Sketch and Create a Diagram strategies to connect the concrete representations created with the unit Squares to visual and symbolic representations in the table. Students should verbalize that the areas represent the number of unit Squares in each figure, while the perimeters represent the linear distance around the outside of each figure. By recording their work in a table, they might be able to begin making generalizations about the relationship between area and perimeter when the area remains the same but the configuration of the square units changes.Notes
Closing Reflection: Unit Squares
How do I use unit Squares in my classroom now? What concepts do I use them to teach? What new ways have I found to use unit Squares to better support student understanding? What are my goals to make unit Squares a more regular part of my instruction?Descriptions of Images and Figures
Back to image The details of the unit Squares and the three figures formed using them are as follows:Three red and two yellow Squares are combined to form the figure of a table; a green square, a red square, a yellow square, and two blue Squares are combined to form the figure of 5 on dice; five green Squares are combined to form a plus symbol.Back to imageThe description below the Squares reads: It took 15 Squares to cover the rectangle. I can add or multiply to get to 15. 5 plus 5 plus 5 equals 15; 3 plus 3 plus 3 plus 3 plus 3 equals 15; 3 times 5 equals 15; 5 times 3 equals 15.Back to image The details of the equations represented by the unit Squares are as follows:- Eight green Squares on the left represent the equations 4 times 2 equals 8
- Four yellow Squares on the right represent the equation 2 times 2 equals 4.
- Both green and yellow Squares together represent the equation 8 plus 4 equals 12.
Back to image The details of three sets of unit square and the respective equations are as follows: In the first set, each row has five blue Squares and a yellow square. This set represents the equation: 3 times 5 and 3 times 1 equals 15 plus 3 equals 18.
- Frank R. Spellman, Nancy E. Whiting(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
Sphere— A container shaped like a ball. Square units— Measurements used to express area (e.g., square feet, square meters, acres). Volume— The capacity of a unit (how much it will hold), measured in cubic units (e.g., cubic feet, cubic meters) or in liquid volume units (e.g., gallons, liters, million gallons). Width— The distance from one side of the tank to the other, measured in linear units. 61 Basic Math Operations 2.11.2 R ELEVANT G EOMETRIC E QUATIONS Circumference C of a circle: C = π d = 2 π r Perimeter P of a square with side a : P = 4 a Perimeter P of a rectangle with sides a and b : P = 2 a + 2 b Perimeter P of a triangle with sides a , b , and c : P = a + b + c Area A of a circle with radius r ( d = 2 r ): A = π d 2 /4 = π r 2 Area A of duct in square feet when d is in inches: A = 0.005454 d 2 Area A of a triangle with base b and height h : A = 0.5 bh Area A of a square with sides a : A = a 2 Area A of a rectangle with sides a and b : A = ab Area A of an ellipse with major axis a and minor axis b : A = π ab Area A of a trapezoid with parallel sides a and b and height h : A = 0.5( a + b ) h Area A of a duct in square feet when d is in inches: A = π d 2 /576 = 0.005454 d 2 Volume V of a sphere with a radius r ( d = 2 r ): V = 1.33 π r 3 = 0.1667 π d 3 Volume V of a cube with sides a : V = a 3 Volume V of a rectangular solid (sides a and b and height c ): V = abc Volume V of a cylinder with a radius r and height H : V = π r 2 h = π d 2 h /4 Volume V of a pyramid: V = 0.33 2.11.3 G EOMETRICAL C ALCULATIONS 2.11.3.1 Perimeter and Circumference On occasion, it may be necessary to determine the distance around grounds or landscapes. To mea-sure the distance around property, buildings, and basin-like structures, it is necessary to determine either perimeter or circumference. The perimeter is the distance around an object; a border or outer boundary. Circumference is the distance around a circle or circular object, such as a clarifier.
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