Mensuration

1. Mensuration T- 1-855-694-8886 Email- [email protected] By iTutor.com

2. Introduction Topic dealing with the use of formulae to calculate Perimeters, Areas and Volumes of plain shapes and solid ones (prisms). Plane: 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. Solid: Geometric figure in three dimensions Surface Area: Total area of all the surfaces of a solid shape or prism. Volume: This is the space occupied by a solid shape or prism. © iTutor . 2000-2013. All Rights Reserved

3. Areas of geometrical shapes Rectangle l w l  w a a squre a  a triangle b h 1/2 b  h h b parallelogram b  h Shape Diagram Area © iTutor . 2000-2013. All Rights Reserved

4. Shape Diagram Area circle  r 2 r Trapezium ½(a +b)h a h b © iTutor . 2000-2013. All Rights Reserved

5. AREA The perimeter of a shape is a measure of distance around the outside. The area of a shape is a measure of the surface/space contained within its perimeter . Area is measured in units 2 Units of distance mm cm m km inches feet yards miles 1 cm 1 cm 2 1 cm 1 cm Metric Imperial Units of area mm 2 cm 2 m 2 km 2 inches 2 feet 2 yards 2 miles 2 Metric Imperial © iTutor . 2000-2013. All Rights Reserved

6. Area of a rectangle Examples To Find the area of a rectangle simply multiply the 2 dimensions together. Area = l x w (or w x l) Find the area of each rectangular shape below. 100 m 50 m 120 m 40 m 1 2 3 4 5 8½ cm 5½ cm 90 feet 50 feet 210 cm 90 cm 5 000 m 2 4500 ft 2 4 800 m 2 46.75 cm 2 18 900 cm 2 © iTutor . 2000-2013. All Rights Reserved

7. Area of a Triangle rectangle area = 2 + 2 triangle area = ½ rectangle area base height Area of a triangle = ½ base x height The area of a triangle = ½ the area of the surrounding rectangle/parallelogram © iTutor . 2000-2013. All Rights Reserved

8. Area of a Triangle Example Find the area of the following triangles. 8 cm 10 cm 14 cm 12 cm 9 cm 16 cm Area = ½ b x h 3.2 m 4.5 m 7 mm 5 mm Area = ½ x 8 x 9 = 36 cm 2 Area = ½ x 10 x 1 = 60 cm 2 Area = ½ x 14 x 16 = 112 cm 2 Area = ½ x 3.2 x 4.5 = 7.2 m 2 Area = ½ x 7 x 5 = 17.5 mm 2 © iTutor . 2000-2013. All Rights Reserved

9. The Area of a Trapezium Area = ( ½ the sum of the parallel sides) x (the perpendicular height) A = ½(a + b)h a b h ½ah ½bh Area = ½ah + ½ bh = ½h(a + b) = ½(a + b)h Find the area of each trapezium 1 8 cm 12 cm 9 cm 2 5 cm 7 cm 6 cm 3 5 cm 3.9 cm 7.1 cm Area = ½ (8 + 12) x 9 = ½ x 20 x 9 = 90 cm 2 Area = ½(7 + 5) x 6 = ½ x 12 x 6 = 36 cm 2 Area = ½(3.9 + 7.1) x 5 = ½ x 11 x 5 = 27.5 cm 2 © iTutor . 2000-2013. All Rights Reserved

10. 32 Sectors Transform Remember C = 2 π r ? ? As the number of sectors   , the transformed shape becomes more and more like a rectangle. What will the dimensions eventually become? ½C r πr A = πr x r = πr 2 The Area of a Circle © iTutor . 2000-2013. All Rights Reserved

11. A =  r 2 A =  x 8 2 A = 201.1 cm 2 A =  r 2 A =  x 10 2 A = 314 cm 2 Find the area of the following circles. A =  r 2 8 cm 1 10 cm 2 Examples © iTutor . 2000-2013. All Rights Reserved

12. Three Dimensional Geometry Three-dimensional figures, or solids, can be made up of flat or curved surfaces. Each flat surface is called a face . An edge is the segment that is the intersection of two faces. A vertex is the point that is the intersection of three or more faces. © iTutor . 2000-2013. All Rights Reserved

13. Boxes A box (also called a right parallelepiped ) is just what the name box suggests. One is shown to the right. A box has six rectangular faces , twelve edges , and eight vertices . A box has a length , width , and height (or base , height , and depth ). These three dimensions are marked in the figure. L W H 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 (cm 3 ) cubic yards, and gallons. The surface area of a three-dimensional object is, as the name suggests, the area of its surface. © iTutor . 2000-2013. All Rights Reserved

14. Volume and Surface Area of a Box The volume of a box is found by multiplying its three dimensions together: L W H Example Find the volume and surface area of the box shown. The volume is The surface area is 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. 8 5 4 © iTutor . 2000-2013. All Rights Reserved

15. Cube A cube is a box with three equal dimensions ( length = width = height). Since a cube is a box, the same formulas for volume and surface area hold. If s denotes the length of an edge of a cube, then its volume is s 3 and its surface area is 6s 2 . A cube is a prism with six square faces. Other prisms and pyramids are named for the shape of their bases. © iTutor . 2000-2013. All Rights Reserved

16. 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. © iTutor . 2000-2013. All Rights Reserved

17. Solution T . S. A. = Area of the 2 triangles + Area of rectangle 1 + Area of rectangle 2 + Area of rectangle 3 = 2 (½ x 6 x 8 ) + (6 x 5.5) + (10 x 5.5) + (8 x 5.5) = ( 2 x 24 ) + 33 + 55 + 44 = 48 + 33 + 55 + 44 Therefore T. S. A. = 180cm² (b ) V = Base area x height = 24 x 5.5 V = 132cm³ 6cm 8cm 5.5cm 10cm A triangular prism has a base in form of a right-angled triangle, with sides 6cm, 8cm and 10cm. If the height of the prism is 5.5cm, sketch the prism and calculate, (a) its total surface area, (b) its volume. Example

18. Cylinders A cylinder is a prism in which the bases are circles. The volume of a cylinder is the area of its base times its height: The surface area of a cylinder is: h r 8cm 3cm Find the surface area of the cylinder. Surface Area = 2 x  x 3(3 + 8) = 6 x 11 = 66 = 207 cm 2 Example © iTutor . 2000-2013. All Rights Reserved

19. 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. © iTutor . 2000-2013. All Rights Reserved

20. Find the volume of the following prisms. 9 m 2 V = 9 x 5 = 45 m 3 8 cm 2 7 mm 2 5 m 4 cm 10 mm 20 mm 2 10 mm 30 m 2 2½ m 40 cm 2 3 ¼ cm V = 8 x 4 = 32 cm 3 V = 7 x 10 = 70 mm 3 V = 20 x 10 = 200 mm 3 V = 30 x 2½ = 75 m 3 V = 40 x 3¼ = 130 cm 3 1 2 3 4 5 6

21. 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: h r © iTutor . 2000-2013. All Rights Reserved

22. 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: r © iTutor . 2000-2013. All Rights Reserved

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