# 1.6. Measure quantities within the least step of the instrument

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Now that we have discussed the units of measurement and the instruments with which to measure, we will look at HOW to measure quantities correctly to within the least step of the instrument.

**Measuring length**

Length is the measurement of something from one end to the other. To measure length, we would use a ruler, e.g.:

How long is this nail? From one end of the nail to the other is 3cm. This is the length of the nail

You measure length all the time. Here are some examples:

- The thickness of some roofing insulation is 370mm
- The length of a pencil is 14cm
- The width of your bedroom is 3m
- The distance from your house to the train station is 3km
- The door is 4 steps away from my chair.
- The door is 3.6 meters away from my chair.

Did you notice that different words such as width or distance can be used to describe the same thing, length?

The standard unit of length in the metric system is the meter. Other units of length and their equivalents in meters are as follows:

- 1 millimetre = 0.001 meter = 1 mm
- 1 centimetre = 0.01 meter = 1 cm
- 1 decimetre = 0.1 meter = 1 dm
- 1 kilometre = 1000 meters = 1 km

We use the following units to measure distance, length, and heights:

- Millimetres (mm); centimetres (cm) ; metres (m) ; kilometres (km)

**Measuring area **

The area of a figure measures the size of the region enclosed by the figure. This is usually expressed in terms of some square unit. A few examples of the units used are square meters, square centimetres, square inches, or square kilometres.

**Area** means the “**floor space**” an object occupies.

We may say that the box below covers 10 tiles on the floor. Therefore its area is 10 tiles.

The **floor area** of box A is:

Length x breadth

If length = 3cm and breadth = 2cm

Area A = 3cm x 2cm = 6cm²

Area of a Square If l is the side-length of a square, the area of the square is l^{2} or l × l. |

Area of a Rectangle The area of a rectangle is the product of its width and length. The formula to calculate area of a rectangle is: A = L x B |

Area of a Parallelogram The area of a parallelogram is b × h, where b is the length of the base of the parallelogram, and h is the corresponding height. To picture this, consider the parallelogram below: We can picture “cutting off” a triangle from one side and “pasting” it onto the other side to form a rectangle with side-lengths b and h. This rectangle has area b × h. |

Area of a TriangleA formula for area of a triangle A = ½ (area of a rectangle on base) A = ½ (l x b) or Consider a triangle with base length b and height h. The area of the triangle is ½ × b × h. To picture this, we could take a second triangle identical to the first, then rotate it and “paste” it to the first triangle as pictured below: or The figure formed is a parallelogram with base length b and height h, and has area b × h. This area is twice that of the triangle, so the triangle has area ½ × b × h. |

Area of a CircleThe area of a circle is Pi × r^{2} or Pi × r × r, where r is the length of its radius. Pi is a number that is approximately 3.14159. |

The **surface area** of a space figure (three-dimensional figure) is the total area of all the faces of the object. Surface area means the area occupied by an object when open up into its components.

- Look at box A below. When opened up, it will look like the B below.

The surface area of the above will therefore be the sum of areas of: Rectangles 1, 2, 3, 4, 5 and 6 respectively.

- What is the surface area of a box whose length is 8, width is 3, and height is 4?

This box has 6 faces: two rectangular faces are 8 by 4, two rectangular faces are 4 by 3, and two rectangular faces are 8 by 3.

Adding the areas of all these faces, we get the surface area of the box:

8 × 4 + 8 × 4 + 4 × 3 + 4 × 3 + 8 × 3 + 8 × 3

= 32 + 32 + 12 + 12 +24 + 24 = 136

Surface Area of a SphereA sphere is a space figure having all of its points the same distance from its centre. The distance from the centre to the surface of the sphere is called its radius. Any cross-section of a sphere is a circle. The surface area S of the sphere is given by the formula S = 4 × pi ×r^{2}. Example: To the nearest tenth, what is the surface area of a sphere having a radius of 4cm?Using an estimate of 3.14 for pi, the surface area would be 4 × 3.14 × 4^{2} = 4 × 3.14 × 4 × 4 = 201 square centimetres. |

Surface Area of a Right PrismThe right prism’s end faces are polygons and the remaining faces are rectangles. A prism is a space figure with two congruent, parallel bases that are polygons. A prism is a (3-dimensional) polyhedron with bases that are parallel, congruent polygons. The sides are parallelograms. We show three examples below. The important features are the areas of the bases and the height, which is the perpendicular distance between the bases. The above left diagram is of a prism with bases that are hexagons. Notice that the slant height, s, is different than the height, h, since the prism is tilted. The prism above on the right has bases that are pentagons and is not tilted. We shall concentrate on prisms such as this which are called right prisms. Pentagonal prism (the bases are pentagons). The greyed lines are edges hidden from view Triangular prism (the bases are triangles). The greyed lines are edges hidden from viewSurface Area = sum of areas of both bases and all sides The area of the bases can be found if we know the dimensions and properties of the polygons which form the bases. The area of all the sides (called the lateral area) can be found if we can measure the base and height (called the slant height and labelled “s”) of each of the parallelograms which form the sides of the prism. Adding the areas of the parallelograms which form the sides can be summarized by multiplying the slant height and the perimeter of a base. We have: Surface Area = sum of areas of both bases and s x P where s is the slant height and P is the perimeter of a base. If the bases are perpendicular to the parallelograms forming the sides (as in a right prism), the slant height s = h. |

Surface Area of a CylinderWe may think of a cylinder as a tin can, with two bases that are circles. The surface wrapping around the circles can be “unwrapped” and shown as a rectangle. A diagram below shows a cylinder on the left. The rectangle at right is the side surface, unwrapped from the cylinder. We know that its length is the circumference of the circular base which is 2πr.The surface area is the sum of the areas of the two circles that are the bases and the area of the rectangle. Since the area of a circle is πr^{2} and the area of a rectangle is lb, we can make a formula for the surface area of a cylinder: S = πr^{2} + 2πrh |