Point, Line Segment, Ray, and Line
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Three Planes
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Two-Dimensional Shapes
Two-dimensional objects only have two dimensions: length and width.
Polygons are two-dimensional shapes made up of line segments. In order to be considered a polygon, a set of line segments must be closed, meaning each line segment meets up with another line segment. Because of this requirement, squares and triangles are considered polygons, while a circle is not a polygon.
Squares are polygons made of four line segments, where each segment is the same length. Rectangles are also made of four line segments, where two parallel segments are equal length, and the other two parallel segments are equal length. Triangles are polygons with three line segments that can be equal length, but don't need to be.
Perimeter and Area
Perimeter is a commonly calculated measurement with two-dimensional shapes in geometry that adds the length of a polygon's line segments. Perimeter calculations are for different applications, including finding how much fencing to put up around your yard.
As illustrated, perimeter calculations are essentially the same for the different shapes: the length of each separate line segment must be added together. For example, if one side of a square, a, measures 12 inches, then using the formula for a square's perimeter, 4a, p is 4 times 12, which equals 48 inches.
The formulas for finding the perimeter of a rectangle and a triangle also require finding the sum of the length of the sides:
Perimeter of a rectangle = (2 * length) + (2 * width)
The perimeter of a triangle = a + b + c
Formula for the perimeter of a square
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Formula for the perimeter of a rectangle
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Formula for the perimeter of a triangle
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The perimeter measurement of a circle is known as circumference. The circumference of a circle is found by using diameter d (the distance across a circle), or radius r (the distance halfway across a circle), and pi, which is a ratio used in geometry that roughly equals 3.14.
Circumference, Diameter, and Radius
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Formula for the circumference of a circle
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Here's an example of how to calculate a circle's circumference. To find the circumference of a circle with a radius r of 4 meters, simply multiply 4 by 2 and by pi (3.14). The circumference of that circle would be approximately 25.12 meters. The circumference = 4 * 2 * 3.14
Area is the measure of the surface of an object. The area of a square, rectangle, triangle, and circle can be found using formulas. Area calculations are used, for instance, when people want to know the square footage of their homes.
The formula for the area of a square is A = l^2 (l = length of a side). The formula for finding the area of a rectangle is A = length * width. When finding the area of a triangle, the formula area = ½ base * height.
Formula for the area of a square
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Formula for the area of a rectangle
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Formula for the area of a triangle
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Formula for the area of a circle
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As an example, to find the area of a triangle with a base b measuring 2 cm and a height h of 9 cm, multiply 1/2 by 2 and 9 to get an area of 9 cm squared. Area is ½ * 2 * 9 = 9.
The formula for the area of a circle A = pi * r^2. This means use 3.14 (for pi) times the radius-squared.
Three-Dimensional Shapes
Unlike two-dimensional objects, three-dimensional objects have a third dimension, depth, and are, therefore, not flat. Cubes, spheres, and pyramids are examples of three-dimensional objects. A cube is an object made of six square sides. A sphere is an object shaped like a ball, where every point on the surface is the same distance from the center of the ball. A cylinder is another three-dimensional object like a can with two circular ends and curved sides.
Surface Area and Volume
When working with three-dimensional objects, formulas are used to find surface area and volume. Surface area is similar to perimeter but instead of adding up the length of the line segments, the areas of each of the shapes composing the three-dimensional object are added together. Knowing this, the formulas for these three-dimensional shapes can be derived. For instance, a cube's surface area is 6 times the area of a single square because it's made of 6 squares.
Surface area can be useful in real life when determining how much paint you need to cover an object. Review the formulas for the surface area of the different shapes:
The formula for the surface area of a cube or a rectangular prism is:
SA = 2lw + 2hw + 2lh. And the formula to use with a cylinder is SA = 2B + Ch (B= Area of the base, C = the circumference). To find the surface area of a sphere, SA = 4 * (pr * r^2).
Formula for surface area of a cube
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Formula for the surface area of a sphere
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Formula for the surface area of a cylinder
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As an example, to find the surface area of a sphere with a radius of 3 feet, simply square the radius and multiply by 4 and by 3.14. The surface area is 113.04 feet squared.
Volume is the amount of a space an object takes up. For a cube, this means finding the area of one square, and finding how much stuff can fit inside if this square is stacked the same number of times as the length (or width). So, when solving for a cube's volume, the length of the side can be multiplied by itself three times because its length, width, and depth are equal.
Volume has many real life uses because it calculates how much an object will hold. For example, you might use the volume of a cylinder to find out how much water your bottle will hold. Even further, you may use the volume of a rectangle to find out how much junk your moving truck can hold when you buy a new house.
The formula to find the volume of a rectangular prism is V = lwh. To find the volume of a sphere, use V = 4/3 * (pi * r^3). Here's an example for finding the volume of a sphere with a radius of a sphere, with a radius of 3 m. Start by cubing the radius to get 27 m squared. Then, multiply 4/3 by pi and 27 to get a final answer of 113.04 m cubed.
And finally, to calculate the volume of a cylinder, use the formula V = Bh. (B = Area of the base)
Formula for volume of a cube
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Formula for the volume of a sphere
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Formula for volume of a cylinder
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Lesson Summary
Geometry is a mathematical subject that deals with shapes and space. Formulas can be used to find the perimeter and area of two-dimensional shapes, such as polygons and circles. Perimeters measure the length of the outside of a two-dimensional object, while area represents the space on the surface of a two-dimensional shape.
In geometry, formulas can also be used to find the surface area and volumes of three-dimensional shapes, like cubes and cylinders. Volume measures the amount of space a three-dimensional object takes up. Surface area measures the area of all sides of a three-dimensional object.
