Sixth Grade Geometry: Concepts and Sample Problems
In sixth grade math, the focus of geometry is on understanding and finding the area of a variety of polygons and the volume of right rectangular prisms. Students also learn to use a coordinate plane to draw polygons, and nets to draw and solve problems about 3dimensional shapes.
Geometry for Sixth Graders
Concepts
Polygons
A polygon is a closed, plane figure with at least three sides; however, you'll often find the term only used for figures of five or more sides. In sixth grade, students will learn how to find the area of triangles, quadrilaterals and polygons.
To figure the area of these various kinds of shapes, students will learn to put them together to make a rectangle or to break them down into simpler shapes, like triangles and rectangles. Then, they can figure the area of each one in the composed or decomposed figure and add them together to find the area of the complete shape.
Right Rectangular Prisms
A right rectangular prism is a 3dimensional figure that has six rectangular sides. Two pairs of those sides (or four sides in all) are composed of opposite parallel faces. The ends (the last pair of sides) are referred to as congruent bases. A cube is a good example.
In fifth grade, students learned to find the volume of right rectangular prisms whose edges are all whole units or numbers (e.g., 2 x 3 x 5). In sixth grade, they'll learn to find the volume of right rectangular prisms where one or more edge length is a fractional number (e.g. 2 1/2 x 3 x 5). Students first fill the prism with unit cubes. In the example given, they would use unit cubes equal to 1/2 unit. After counting the cubes used, students will realize that another way to calculate volume is by multiplying all three sides.
Coordinate Planes and Polygons
Your child learned in fifth grade to use coordinate planes and to graph ordered pairs on it. This year, he'll learn to graph a number of ordered pairs on it and draw a polygon. He'll learn to figure the lengths of each horizontal and vertical side of the polygon.
Nets
Nets are 2dimensional plane figures that can be folded into 3dimensional geometric solids. In sixth grade, your child will learn to use nets to figure the surface area of different 3dimensional figures.
Problems
In sixth grade, students are taught how geometry can be useful in real life. Below, for each of the four areas you'll find a sample problems for the concept and then a practical application.
Polygons
1. Find the area of this trapezoid where the base is eight inches, the top is four inches and the height is three inches.
 Break the trapezoid into simpler shapes. Draw perpendicular lines from the top to the base so there is now a 3 x 4 rectangle in the middle, with a triangle on each side, both of which have a base of 2 and a height of 3.
Since the two triangles are the same size, they can be combined into one 2 x 3 rectangle.
Perform the calculations to find the area of the two rectangles:
 3 x 4 = 12
2 x 3 = 6
Add the areas of both rectangles together to get the area of the trapezoid.
 12 + 6 = 18
2. Mrs. Smith installed a 15' x 30' swimming pool. She wants to add a sidewalk around the pool that is 5' wide. The cement company needs to know how many square feet the sidewalk is so they can figure how much cement to bring. What will Mrs. Smith tell them?
 To figure this problem, your sixth grader might draw a chart on graph paper of the pool with the sidewalk around it. They would then have a 15' x 30' rectangle, with a 25' x 40' around it. First, calculate the area of each rectangle and subtract the smaller one from the larger one to get the area of the sidewalk. Therefore:
 15' x 30' = 450 sq. ft.
25' x 40' = 1000 sq. ft.
1000 sq. ft.  450 sq. ft. = 550 sq. ft, which is what Mrs. Smith needs to tell the cement company.
Right Rectangular Prisms
1. Find the volume of a prism with the dimensions 4 by 5 by 6.
 The formula is Volume = Length x Width x Height, or V = lwh.
 V = 4 x 5 x 6 = 120 cubic units
2. The trailer of a semitruck is a right rectangular prism, with a height of 9 feet. A sewing machine company wants to ship items in a truck trailer that is 8 feet wide and 16 feet long. The boxes are 2' x 2' x 2'. How many boxes of sewing machines can they ship in the trailer?
 Some children will need to draw a picture to solve this problem. They can put 4 boxes across the width, 8 boxes from front to back and 4 boxes high.
 4 x 8 x 4 = 128 boxes
Coordinate Planes and Polygons
1. Graph the ordered pairs below and join the points to make a polygon.
 (3, 8)
(3, 12)
(7, 15)
(11, 12)
(11, 8)
(3, 8)
 Since the first and last coordinates are the same, this is a closed figure, making it a polygon with five sides.
2. Captain Jones, the pilot of a plane traveling from New York to Paris, asks his navigator how far they are from Paris. The navigator sees that Paris is located at N48 degrees, E2 degrees. Help him out by charting that on a coordinate plane.
 Begin by placing a point at (2, 48).
The navigator finds the location of the plane to be at approximately N48 degrees, W15 degrees. Help him out by charting this point on the same coordinate plane.
 Place another point at (15, 48).
How far is the airplane from Paris?
 Count the points from 15 to 2, which is a distance of 17 degrees.
Nets
1. The dimensions of a right rectangular prism are 6 x 5 x 4. What is the surface area of the prism?
 The formula for the surface area of a right rectangular prism is Surface Area = 2(lw) + 2(hl) + 2(hw). The work for this problem would be:
 SA = 2(6 x 5) + 2(4 x 6) + 2(4 x 5)
SA = 2(30) + 2(24) + 2(20)
SA = 60 + 48 + 40
SA = 148 sq. units
2. A contractor is building a house that is in the shape of a right rectangular prism with outside dimensions of 50 feet long, 30 feet wide and 10 feet high. He needs to know the surface area of the house so he can figure how much paint to order. What would the surface area of the house be?
 Draw a net for the house, perhaps even cut it out and make the prism. Exclude the areas of the top and bottom since they won't need paint. Of the four remaining sides, two are 50' x 10' and two are 30' x 10'.
 2 (50 x 10) = 1000 sq. ft.
2 (30 x 10) = 600 sq. ft.
1000 + 600 = 1600 sq. ft.
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