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Circles

Circles. "A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another. And the point is called the center of the circle.". Euclid wrote:.

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Circles

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  1. Circles

  2. "A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another. And the point is called the center of the circle." Euclid wrote:

  3. The diameter of a circle is the length of a line segment whose endpoints lie on the circle and which passes through the centre of the circle. This is the largest distance between any two points on the circle. The diameter of a circle is twice its radius. Parts of the circle

  4. The circumference of a circle is the length around it. The circumference of a circle can be calculated from its diameter using the formula: C = πd Or, substituting the radius for the diameter: C = 2πr The circumference of a circle

  5. When a circle's diameter is 1 unit, its circumference is pi units • In other words, the ratio of a circle’s circumference to its diameter is π

  6. As proved by Archimedes, the area enclosed by a circle is π multiplied by the radius squared: A = πr2 The area of a circle

  7. Prisms & Solids (polyhedra)

  8. Prisms • In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy (not in the same plane as the first), and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases. • Prisms are named for their base, so a prism with a pentagonal base is called a pentagonal prism. • How is the prism to the right called?

  9. Volume • The volume of a prism is the product of the area of the base and the height. • The volume is then: V = Bh, where B is the area of the base and h is the height of the prism. (That’s how we get V = lwh.) • In the case of a non-right prism, note that this means the perpendicular distance.

  10. Formulas for volume

  11. Calculate the volume

  12. Calculate the volume

  13. Surface Area • Surface area is the measure of how much exposed area a solid object has, expressed in square units. • For polyhedra (objects with flat polygonal faces) the surface area is the sum of the areas of its faces.

  14. Formulas for surface are

  15. Calculate the surface area

  16. Calculate the surface area

  17. Cones and Pyramids

  18. A cone is a three-dimensionalgeometric shape that tapers smoothly from a flat, usually circular base to a point called the apex or vertex. Cones

  19. In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. Pyramids l

  20. Volume of cones and pyramids

  21. Calculate the volume

  22. Calculate the volume

  23. The surface area of a cone is ½(2πr)l + B, where l is the slant height and B is the area of the base. Surface Area of a Cone

  24. The surface area of a pyramid is ½Pl + B, where P is the perimeter of the base, l is the slant height and B is the area of the base. Surface Area of cones and Pyramids l

  25. Volume of cones and pyramids

  26. Calculate the surface area

  27. Calculating the area of regular polygons with 5 or more sides • SCENARIO Have you ever noticed that every stop sign looks exactly the same, every yield sign looks exactly the same, and so on? • This is because the Federal Highway Administration has standards that indicate the exact sizes and colors of roadway signs. Most of the sign shapes are polygons.

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