7th Grade Chapter 10
10-1 Area: Parallelograms
To find the area of a parallelogram, you can cut off a right triangle from one end and move it to the opposite end to form a rectangle. Because of this, the area formulas for rectangles and parallelograms are the same.
10-2 Area: Triangles and Trapezoids
Students know how to find the area of a parallelogram. In this lesson, students adapt the formula for the area of a parallelogram to find the areas of triangles and trapezoids. A triangle can be viewed as half of a parallelogram that has the same base and height as the triangle. A trapezoid can be viewed as two triangles with the same height, but different bases.
10-3 Area: Circles
The formula for area of a circle is A=pi * r squared. Since pi is irrational and a decimal that never terminates, the exact area of a circle must be written with pi shown as a factor. An approximate area of a circle can be found by using a decimal approximation, such as 3.14 or 22/7.
10-4 Space Figures
A space figure is a three-dimensional figure. A net is a two-dimensional pattern for a three-dimensional figure. To help identify the type of figure that is modeled by a net, look at the shapes in the net.
10-5 Surface Area: Prisms and Cylinders
The bases of prisms are various polygons. You cannot use a single formula for the areas of the bases of all prisms. Therefore, B is used to represent “Area of the base.” The bases of cylinders are always circles. Therefore you can use the formula S.A. = 2 * pi * r * h + 2 * pi * r squared for the surface area of a cylinder. However, it is easier to remember just one formula for surface area of either a prism or cylinder. S.A. = L.A + 2B
10-6 Surface Area: Pyramids, Cones and Spheres
Note the similarity between the formulas for area of a rectangle and lateral areas of prisms and cylinders:
A = bh and L.A. = pl
(p is perimeter or circumference)
Note also the similarity between the formulas for area of a triangle and lateral areas of pyramids and cones:
A = 1/2 bh and L.A. = 1/2 pl
Theses are handy analogies that can help you remember not only the lateral-area formulas, but also the surface-area formulas, each of which involves adding lateral area and base areas.
10-7 Volume: Prisms and Cylinders
As in formulas for surface areas, you can use B to represent the area of a base in formulas for volumes. The formula
V = Bh covers all of the possibilities of base shapes for prisms and cylinders.
10-9 Volume of Pyramids, Cones and Spheres
It is important to note that the formula for the volumes of cones and pyramids uses each figure’s height, not slant height. The height of a cone or pyramid is the length of the segment from the vertex perpendicular to the base. An interesting experiment to convince students that the formula for the volume of a cone makes sense is to have them cut two figures out of paper: a cone and a cylinder that have exactly the same height and circumference. Then ask students to fill the cone with rice and pour the rice into the cylinder. Most students are surprised to discover that it takes three cones of rice to fill the cylinder.
10-10 Scale Factors and Solids
Similar solids have the same shape and proportional corresponding measurements. The idea that doubling linear dimensions does not simply double the area and the volume is counter-intuitive for most people. Experiments with models and actual measurements help make the effect of changing dimensions seem more reasonable.
To find the area of a parallelogram, you can cut off a right triangle from one end and move it to the opposite end to form a rectangle. Because of this, the area formulas for rectangles and parallelograms are the same.
10-2 Area: Triangles and Trapezoids
Students know how to find the area of a parallelogram. In this lesson, students adapt the formula for the area of a parallelogram to find the areas of triangles and trapezoids. A triangle can be viewed as half of a parallelogram that has the same base and height as the triangle. A trapezoid can be viewed as two triangles with the same height, but different bases.
10-3 Area: Circles
The formula for area of a circle is A=pi * r squared. Since pi is irrational and a decimal that never terminates, the exact area of a circle must be written with pi shown as a factor. An approximate area of a circle can be found by using a decimal approximation, such as 3.14 or 22/7.
10-4 Space Figures
A space figure is a three-dimensional figure. A net is a two-dimensional pattern for a three-dimensional figure. To help identify the type of figure that is modeled by a net, look at the shapes in the net.
10-5 Surface Area: Prisms and Cylinders
The bases of prisms are various polygons. You cannot use a single formula for the areas of the bases of all prisms. Therefore, B is used to represent “Area of the base.” The bases of cylinders are always circles. Therefore you can use the formula S.A. = 2 * pi * r * h + 2 * pi * r squared for the surface area of a cylinder. However, it is easier to remember just one formula for surface area of either a prism or cylinder. S.A. = L.A + 2B
10-6 Surface Area: Pyramids, Cones and Spheres
Note the similarity between the formulas for area of a rectangle and lateral areas of prisms and cylinders:
A = bh and L.A. = pl
(p is perimeter or circumference)
Note also the similarity between the formulas for area of a triangle and lateral areas of pyramids and cones:
A = 1/2 bh and L.A. = 1/2 pl
Theses are handy analogies that can help you remember not only the lateral-area formulas, but also the surface-area formulas, each of which involves adding lateral area and base areas.
10-7 Volume: Prisms and Cylinders
As in formulas for surface areas, you can use B to represent the area of a base in formulas for volumes. The formula
V = Bh covers all of the possibilities of base shapes for prisms and cylinders.
10-9 Volume of Pyramids, Cones and Spheres
It is important to note that the formula for the volumes of cones and pyramids uses each figure’s height, not slant height. The height of a cone or pyramid is the length of the segment from the vertex perpendicular to the base. An interesting experiment to convince students that the formula for the volume of a cone makes sense is to have them cut two figures out of paper: a cone and a cylinder that have exactly the same height and circumference. Then ask students to fill the cone with rice and pour the rice into the cylinder. Most students are surprised to discover that it takes three cones of rice to fill the cylinder.
10-10 Scale Factors and Solids
Similar solids have the same shape and proportional corresponding measurements. The idea that doubling linear dimensions does not simply double the area and the volume is counter-intuitive for most people. Experiments with models and actual measurements help make the effect of changing dimensions seem more reasonable.