Houghton Mifflin Mathematics Grade 4: Chapter 10: Finding Area and Volume: When Students Ask


Finding Area and Volume Introducing the Concept Developing the Concept Worksheets Common Polygons

Perimeter, Area, and Volume

Perimeter, area, and volume are measurements of geometric figures. Geometric figures can be defined by their attributes and properties.

Geometric Figures

Figure ABCD below is a quadrilateral. A quadrilateral is a polygon (a flat, closed, plane figure made up of three or more line segments called sides). A quadrilateral has four sides.

A point is an exact location in space.
A line can be drawn through any two points and goes on without end in both directions.
$AB$ , $AC$ , $BD$ , and $CD$ are line segments. A line segment is part of a line. It has two endpoints.

In the figure below, sides $AB$ and $CD$ are parallel. Parallel lines are always the same distance apart. Sides $AC$ and $CD$ are perpendicular to each other. Perpendicular lines are two lines that intersect to form a right angle.

$AB$ is parallel to $CD$ .
$AC$ is perpendicular to $CD$ .
$BD$ meets $CD$ at point D. These lines are called intersecting lines.

An angle is formed by two rays with a common endpoint. The common endpoint is the vertex.

A ray is part of a line having one endpoint and continuing without end in one direction.
$ray DB$ and $DC$ form $times$ BDC

This quadrilateral has four angles. The figure contains two right angles, one obtuse angle, and one acute angle. Angles are classified by their measure. A right angle forms a square corner. An obtuse angle is greater than a right angle. An acute angle is less than a right angle.

Some plane figures are symmetric. This quadrilateral has no line of symmetry. It cannot be folded so that two parts match exactly.

This circle has symmetry. The circle can be folded along the diameter. The diameter is a line of symmetry. When the circle is folded along the diameter, the circle is divided into two parts that match exactly. These matching parts are congruent. Figures are congruent if they have the same size and shape.

See the Common Polygons worksheet for the names, characteristics, and classification of common polygons. Also see Polygons and Solid Figures.

Measuring Geometric Figures

The distance around a plane figure can be measured. This distance is the perimeter of the figure. To find the perimeter of a polygon, add the lengths of the sides.

The perimeter is 18 feet.

Finding area is another way to measure a figure. Area is the number of square units needed to cover a figure. To find the area of a rectangle, count the number of square units or use a formula. Multiply the length (l) by the width (w) to find the area (A) of a rectangle.

To find the area of a complex figure, separate it into simpler figures, find the areas, and then add the areas together.

Figures with the same perimeters can have different areas.

Figures with the same area can have different perimeters.

Volume is the number of cubic units that make up a solid figure. Different kinds of solid figures are shown below.

The volume of a rectangular prism can be found by counting the number of cubic units or by using a formula. The formula for finding the volume of a rectangular prism is V = l $times$ w $times$ h.

The volume of this rectangular prism is 30 cubic meters.

Another way to measure a solid figure is to find the surface area. Surface area is the sum of the areas of all the faces of the solid figure. Since opposite faces of rectangles or cubes have the same area, you can also multiply each area by 2 and then find the sum of the areas of the faces.

The surface area of this rectangular prism is 52 yd².

Perimeter, area, volume, and surface area are measurements of geometric figures. Perimeter and area are measurements of plane figures and surface area and volume are measurements of solid figures.