# 8.6 - Volume and Surface Area

8.6  The student will

a)  solve problems, including practical problems, involving volume and surface area of cones and square-based pyramids;

b)  describe how changing one measured attribute of a rectangular prism affects the volume and surface area.

### BIG IDEAS

• I can order the square feet of canvas required to make a tent, determine how many jelly beans fill a canister, figure how much paint will cover walls, ceiling and floor of a room, and know how much water will fill a fish aquarium.
• I will be able to find volume and surface area as appropriate in various situations.

### UNDERSTANDING THE STANDARD

2016 VDOE Curriculum Framework - 8.6 Understanding

·  A polyhedron is a solid figure whose faces are all polygons.

·  Nets are two-dimensional representations of a three-dimensional figure that can be folded into a model of the three-dimensional figure.

·  Surface area of a solid figure is the sum of the areas of the surfaces of the figure.

·  Volume is the amount a container holds.

·  A rectangular prism is a polyhedron that has a congruent pair of parallel rectangular bases and four faces that are rectangles. A rectangular prism has eight vertices and twelve edges. In this course, prisms are limited to right prisms with bases that are rectangles.

·  The surface area of a rectangular prism is the sum of the areas of the faces and bases, found by using the formula S.A. = 2lw + 2lh + 2wh. All six faces are rectangles.

·  The volume of a rectangular prism is calculated by multiplying the length, width and height of the prism or by using the formula V = lwh.

·  A cube is a rectangular prism with six congruent, square faces. All edges are the same length.
A cube has eight vertices and twelve edges.

·  A cone is a solid figure formed by a face called a base that is joined to a vertex (apex) by a curved surface. In this grade level, cones are limited to right circular cones.

·  The surface area of a right circular cone is found by using the formula, S.A. = pr2 + prl, where l represents the slant height of the cone. The area of the base of a circular cone is pr2.

·  The volume of a cone is found by using V = pr2h, where h is the height and pr2 is the area of the base.

·  A square-based pyramid is a polyhedron with a square base and four faces that are triangles with a common vertex (apex) above the base. In this grade level, pyramids are limited to right regular pyramids with a square base.

·  The volume of a pyramid is  Bh, where B is the area of the base and h is the height.

·  The surface area of a pyramid is the sum of the areas of the triangular faces and the area of the base, found by using the formula S.A. = lp + B where l is the slant height, p is the perimeter of the base and B is the area of the base.

·  The volume of a pyramid is found by using the formula V = Bh, where B is the area of the base and h is the height.

·  The volume of prisms can be found by determining the area of the base and multiplying that by the height.

·  The formula for determining the volume of cones and cylinders are similar. For cones, you are determining  of the volume of the cylinder with the same size base and height. The volume of a cone is found by using V = pr2h. The volume of a cylinder is the area of the base of the cylinder multiplied by the height, found by using the formula, V= pr2h, where h is the height and pr2 is the area of the base.

·  The calculation of determining surface area and volume may vary depending upon the approximation for pi. Common approximations for π include 3.14, , or the pi button on the calculator.

·  When the measurement of one attribute of a rectangular prism is changed through multiplication or division the volume increases by the same factor by which the attribute increased. For example, if a prism has a volume of 2· 3· 4, the volume is 24 cubic units. However, if one of the attributes is doubled, the volume doubles. That is, 2· 3· 8, the volume is 48 cubic units or 24 doubled.

·  When one attribute of a rectangular prism is changed through multiplication or division, the surface area is affected differently than the volume. The formula for surface area of a rectangular prism is 2(lw) + 2(lh) + 2(wh) when the width is doubled then four faces are affected. For example, a rectangular prism with length = 7 in., width = 4 in., and height = 3 in. would have a surface area of or 122 square inches. If the height is doubled to 6 inches then the surface area would be found by  or 188 square inches.

### ESSENTIALS

• How does the volume of a three-dimensional figure differ from its surface area?
Volume is the amount a container holds.
Surface area of a figure is the sum of the area on surfaces of the figure.
• How are the formulas for the volume of prisms and cylinders similar?
For both formulas you are finding the area of the base and multiplying that by the height.
• How are the formulas for the volume of cones and pyramids similar?
For cones you are finding 1/3 of the volume of the cylinder with the same size base and height.
For pyramids you are finding 1/3 of the volume of the prism with the same size base and height.
• In general what effect does changing one attribute of a prism by a scale factor have on the volume of the prism?
When you increase or decrease the length, width or height of a prism by a factor greater than 1, the volume of the prism is also increased by that factor.

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

·  8.6a1  Distinguish between situations that are applications of surface area and those that are applications of volume.

·  8.6a2  Determine the surface area of cones and square-based pyramids by using concrete objects, nets, diagrams and formulas.

·  8.6a3  Determine the volume of cones and square-based pyramids, using concrete objects, diagrams, and formulas.

·  8.6a4  Solve practical problems involving volume and surface area of cones and square-based pyramids.

·  8.6b1  Describe how the volume of a rectangular prism is affected when one measured attribute is multiplied by a factor of , , , 2, 3, or 4.

·  8.6b2  Describe how the surface area of a rectangular prism is affected when one measured attribute is multiplied by a factor of  or 2.

### KEY VOCABULARY

surface area, volume, square pyramid, square pyramid, triangular pyramid, concrete objects, nets, diagrams, cone, right cylinder, rectangular prism, attribute, vertex, edge, face, base, diagonals

Updated: Nov 20, 2018