Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right rectangular prisms.

Arizona Academic Standards:
7.G.B.6

Solve mathematical problems and problems in a real-world context involving area of two-dimensional objects composed of triangles, quadrilaterals, and other polygons. Solve mathematical problems and problems in real-world context involving volume and surface area of three-dimensional objects composed of cubes and right prisms.

Common Core State Standards:
Math.7.G.6 or 7.G.B.6

Tennessee Academic Standards:
7.G.B.5

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Georgia Standards of Excellence (GSE):
7.GSR.5.6

Solve realistic problems involving surface area of right prisms and cylinders.

Louisiana Academic Standards:
7.G.B.6

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (Pyramids limited to surface area only.)

North Carolina - Standard Course of Study:
7.G.6

Solve real-world and mathematical problems involving:

Area and perimeter of two-dimensional objects composed of triangles, quadrilaterals, and polygons.

Volume and surface area of pyramids, prisms, or three-dimensional objects composed of cubes, pyramids, and right prisms.

New York State Next Generation Learning Standards:
7.G.6

Solve real-world and mathematical problems involving area of two-dimensional objects composed of triangles and trapezoids. Note:The inclusive definition of a trapezoid will be utilized, which defines a trapezoid as "A quadrilateral with at least one pair of parallel sides." (This definition includes parallelograms and rectangles.) Solve surface area problems involving right prisms and right pyramids composed of triangles and
trapezoids. Note: Right prisms include cubes. Find the volume of right triangular prisms, and solve volume problems involving three-dimensional objects composed of right rectangular prisms.

Pennsylvania Core Standards:
CC.2.3.7.A.1

Solve real-world and mathematical problems involving angle measure, area, surface area, circumference, and volume.

Pennsylvania Core Standards:
M07.C-G.2.2.2

Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Arkansas Academic Standards:
7.GM.3

Apply the formulas for the volume and surface area of right rectangular prisms, rectangular pyramids, triangular prisms, and triangular pyramids to solve real-world and mathematical problems.

7th Grade Math - Area, Surface Area, & Volume Lesson

The area of a figure can be found by decomposing it into smaller, non-overlapping figures.

The area of the large figure is equal to the sum of the areas of the smaller figures.

Example:

Start by decomposing the trapezoid into 2 congruent triangles and a rectangle.

Next, find the area of one of the triangles and the rectangle.

Area of Rectangle

=

length × width

=

13 cm × 12 cm

=

156 sq cm

Area of Triangle

=

$\frac{1}{2}$× base × height

=

$\frac{1}{2}$× 5 cm × 12 cm

=

30 sq cm

Then, find the sum of the 3 areas.

156 sq cm + 30 sq cm + 30 sq cm = 216 sq cm

So, the area of the trapezoid is 216 square centimeters.

The surface area of a prism can be found by adding the areas of its faces.

Example:

A rectangular prism has six rectangular faces. The prism has three pairs of congruent faces.

Two of the faces have dimensions of 5 inches by 10 inches. Find the area of one of the faces.

5 in. × 10 in. = 50 sq in.

The area of one face is 50 square inches, so the combined area of both faces is 2 × 50 sq in. = 100 sq in.

Two of the faces have dimensions of 5 inches by 12 inches. So, the area of one of the faces is 5 in. × 12 in. = 60 sq in. The combined area of both faces is 2 × 60 sq in. = 120 sq in.

Two of the faces have dimensions of 10 inches by 12 inches. So, the area of one of the faces is 10 in. × 12 in. = 120 sq in. The combined area of both faces is 2 × 120 sq in. = 240 sq in.

Next, find the sum of the areas.

100 sq in. + 120 sq in. + 240 sq in. = 460 sq in.

So, the surface area of the rectangular prism is 460 square inches.

The volume of a prism can be calculated by multiplying the area of its base, B, by its height, h.

Volume = Bh

Example:

The volume of a prism can be calculated by multiplying the area of the base, B, times the height, h.

Volume

=

Bh

=

(122.1 sq cm)(16 cm)

=

1,953.6 cu cm

So, the volume of the prism is 1,953.6 cubic centimeters.