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Geometric Shapes and Formulas. Planes. A plane is a flat surface (think tabletop) that extends forever in all directions. It is a two-dimensional figure. Three non-collinear points determine a plane. So far, all of the geometry we’ve done in these lessons took place in a plane.
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Planes • A plane is a flat surface (think tabletop) that extends forever in all directions. • It is a two-dimensional figure. • Three non-collinear points determine a plane. • So far, all of the geometry we’ve done in these lessons took place in a plane. • But objects in the real world are three-dimensional, so we will have to leave the plane and talk about objects like spheres, boxes, cones, and cylinders.
Perimeter is the distance around a figure. • There are three ways to find the perimeter: • When the lengths of the sides are different, add the lengths of the sides. (S + S + S…=P) • When two pairs of sides are the same measurement, multiply 2 times the length and 2 times the width. Then add the products. (2xL) + (2xW)=P • When all the sides are the same length, multiply the number of sides times the length of one side. (SxL=P) • Ready for some problems?
2 L x W = units 2 3cm x 12cm = 36cm 3cm 12cm 7mm x 7mm = 49mm 2 7mm Area is the number of square units on a flat surface. To find the surface area of a quadrilateral use a formula.
Different Names/Same idea • Length x Width = Area • Side x Side = Area • Base x Height = Area Free powerpoint template: www.brainybetty.com
REAL formula for area of squares • Area= S^2 • Area= Side x Side (side squared) (just a different way of saying length x width) Side one Side two Free powerpoint template: www.brainybetty.com
Parallelogram Area • Same idea as squares & rectangles, but they change the words to Base ( length: bottom of shape) and height (width) Free powerpoint template: www.brainybetty.com
Parallelogram Area • Area=Base x Height • (Area=length x width) The diagonal line is NOT the height!!! 5 Height (width) 8 BASE (length) Base 8 x Height 5 = Area 40 Free powerpoint template: www.brainybetty.com
Area of a Triangle • ½ Base x Height = Area • (It’s ½ because ½ of the “square” is missing) Height 5 Base 8 Height ½ Base x Height = Area ½ (8) x 5 = Area 4 x 5 = 20 Base Free powerpoint template: www.brainybetty.com
Volume and Surface Area • The volume of a three-dimensional object measures the amount of “space” the object takes up. • Volume can be thought of as a capacity and units for volume include cubic centimeters cubic yards, and gallons. • The surface area of a three-dimensional object is, as the name suggests, the area of its surface.
H W L Volume and Surface Area of a Box • The volume of a box is found by multiplying its three dimensions together: • The surface area of a box is found by adding the areas of its six rectangular faces. Since we already know how to find the area of a rectangle, no formula is necessary.
4 5 8 Example • Find the volume and surface area of the box shown. • The volume is • The surface area is
Prisms • A prism is a three-dimensional solid with two congruent bases that lie in parallel planes, one directly above the other, and with edges connecting the corresponding vertices of the bases. • The bases can be any shape and the name of the prism is based on the name of the bases. • For example, the prism shown at right is a triangular prism. • The volume of a prism is found by multiplying the area of its base by its height. • The surface area of a prism is found by adding the areas of all of its polygonal faces including its bases.
Pyramids • A pyramid is a three-dimensional solid with one polygonal base and with line segments connecting the vertices of the base to a single point somewhere above the base. • There are different kinds of pyramids depending on what shape the base is. To the right is a rectangular pyramid. • To find the volume of a pyramid, multiply one-third the area of its base by its height. • To find the surface area of a pyramid, add the areas of all of its faces.
h r Cones • A cone is like a pyramid but with a circular base instead of a polygonal base. • The volume of a cone is one-third the area of its base times its height: • The surface area of a cone is:
r Spheres • Sphere is the mathematical word for “ball.” It is the set of all points in space a fixed distance from a given point called the center of the sphere. • A sphere has a radius and diameter, just like a circle does. • The volume of a sphere is: • The surface area of a sphere is: