Presentation on theme: "Emily Reverman. In this portfolio, you will see how to develop formulas for the area of different shapes (rectangle, parallelogram, trapezoid, and a."— Presentation transcript:
In this portfolio, you will see how to develop formulas for the area of different shapes (rectangle, parallelogram, trapezoid, and a circle), as well as how to develop formula for the volume of different shapes, including cylinder, cone, and pyramid.
Area is the number of square units in a two dimensional shape. A two dimensional shape is a shape that only has two components-length and width-and stays on the same plane.
A rectangle is a four sided figure with four straight sides and four right angles.
Count how many squares cover the region of the rectangle. There is are total of 34 squares inside the rectangle. This means the area of the rectangle is 34 square units. If you count the total number of squares in each row there are 9 and there are 4 squares in each column. To find the total number of squares you can multiply 9 by 4. 9x4=34 Length=9 Width=4 Length x width = area of rectangle
A parallelogram is a four sided figure with opposite sides being parallel. Parallel means that the two lines will continue on the same plane, in a straight line, and will never cross each other.
When looking at the parallelogram to the right, notice that if you remove the triangle on the right, you can place it on the left side of the shape and create a rectangle. 4cm Since you can create a rectangle by rearranging the shape, the area of the parallelogram is the same of a rectangle.
A parallelogram can always be changed into a rectangle with the same base, the same height, and the same area. Therefore, the formula for the parallelogram is the same as the rectangle. Area= Length X Width
A triangle is a figure with three straight sides and three angles
If two equal triangles are put together, they will form a rectangle. Therefore, the area of a triangle is ½ the area of a rectangle. A= ½ bh
A trapezoid is a quadrilateral with only one pair of parallel sides.
One way to find the area of a trapezoid is to brake the trapezoid up into smaller, more known shapes, such as triangles and rectangles. Area of triangle + area of rectangle = area of trapezoid ½ (12 x 5) + (12 x 9) = 138 ½ bh + bh = area of a trapezoid ½ h (b1 + b2) = area of a trapezoid
You can also divide the trapezoid into two triangles and determine the formula for area that way. Area or triangle 1 + area of triangle 2= area of trapezoid ½ bh + (1/2)bh ½ h (b1 + b2)
Circumference: The perimeter, or the distance, around the outer edge of the circle. ◦ C= 2πr Radius: The distance from the center of the circle to the outside edge Diameter: The longest distance across the circle. It cuts through the center of the circle, making it the longest distance. Pi: The ratio of a circle’s circumference to its diameter.
When given a circle, you can cut it into sections and arrange them together to make them look like a parallelogram. The formula for the area of a parallelogram is A= bh The base of the parallelogram above would be equal to half the circumference of the circle. B= C½ We know that C= 2πr so… B=πr The height of the parallelogram would be the radius. The area of the circle would be A= πr^2
Volume is the amount of space that a substance or object occupies, or that is enclosed within a container.
A cylinder is a solid with two congruent parallel bases and sides with parallel elements that join corresponding points on the bases.
Volume of cylinder = area of the base(height) In this example, the base is a circle. The formula for area of a circle is A=πr^2. So the volume of this cylinder is V= π(r^2)h
A cone is a solid or hollow object that tapers from a circular or roughly circular base to a point.
The formula for volume of a cone can be found by comparing it to a cylinder with the same base. Fill the cylinder with water and see how many of the cones it takes to fill the cylinder. If the base is the same, it will take three cones to fill up the cylinder. Since volume of cylinder = 3(volume of cone) Volume of cone = (1/3) volume of cylinder We know that the formula for the volume of a cylinder is area of the base times the height. In this care it would be π(r^2)h Volume of cone = (1/3)π(r^2)h
A pyramid is a structure with a square or triangular base and sloping sides that meet in a point at the top.
The volume of a pyramid can be found by comparing it to a cylinder with the same base and height in the same way we found the volume of a cone. It takes three pyramids to fill a cylinder Volume of cylinder = 3 (volume of pyramid) (area of base)h = 3 (volume of pyramid) So for this particular pyramid… ◦ bwh = 3 (volume of pyramid) ◦ Volume of pyramid = (1/3)bwh