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PART II.

ON AREAS.

DEFINITIONS.

1. The altitude (or height) of a parallelogram with reference to a given side as base, is the perpendicular distance between the base and the opposite side.

2. The altitude (or height) of a triangle with reference to a given side as base, is the perpendicular distance of the opposite vertex from the base.

NOTE. It is clear that parallelograms or triangles which are between the same parallels have the same altitude.

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3. The area of a figure is the amount of surface contained within its bounding lines.

4. A square inch is the area of a square drawn on a side one inch in length.

Square
inch

5. Similarly a square centimetre is the area of a square drawn on a side one centimetre in length.

Sq.

cm.

The terms square yard, square foot, square metre are to be understood in the same sense.

6. Thus the unit of area is the area of a square on a side of unit length.

THEOREM 23.

Area of a rectangle. If the number of units in the length of a rectangle is multiplied by the number of units in its breadth, the product gives the number of square units in the area.

C

A

Let ABCD represent a rectangle whose length AB is 5 feet, and whose breadth AD is 4 feet.

Divide AB into 5 equal parts, and BC into 4 equal parts, and through the points of division of each line draw parallels to the other.

The rectangle ABCD is now divided into compartments, each of which represents one square foot.

Now there are 4 rows, each containing 5 squares,

.. the rectangle contains 5 × 4 square feet.

Similarly, if the length = a linear units, and the breadth = b linear units

the rectangle contains ab units of area. And if each side of a square = a linear units, the square contains a2 units of area.

These statements may be thus abridged:

the area of a rectangle = length × breadth..
the area of a square = (side)2

..(1),

..(ii).

Q. E.D.

COROLLARIES. (i) Rectangles which have equal lengths and equal breadths have equal areas.

(ii) Rectangles which have equal areas and equal lengths have also equal breadths.

NOTATION.

The rectangle ABCD is said to be contained by AB, AD; for these adjacent sides fix its size and shape.

A rectangle whose adjacent sides are AB, AD is denoted by rect. AB, AD, or simply AB × AD.

A square drawn on the side AB is denoted by sq. on AB, or AB2.

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2. Draw a figure to shew that the square on a straight line is four times the square on half the line.

3. Use squared paper to shew that the square on 1"-102 times the square on 0.1".

4. If 1" represents 5 miles, what does an area of 6 square inches represent?

EXTENSION OF THEOREM 23.

The proof of Theorem 23 here given supposes that the length and breadth of the given rectangle are expressed by whole numbers; but the formula holds good when the length and breadth are fractional.

This may be illustrated thus:

Suppose the length and breadth are 3-2 cm. and 24 cm.; we shall shew that the area is (3.2 × 2·4) sq. cm.

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EXERCISES.

(On the Area of a Rectangle.)

Draw on squared paper the rectangles of which the length (a) and breadth (b) are given below. Calculate the areas, and verify by the

actual counting of squares.

1. a=2", b=3".

3. a=0.8", b=3.5′′.

5. a=2.2", b=1.5′′.

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Calculate the areas of the rectangles in which

7. a=18 metres, b=11 metres. 8. a 7 ft., b=72 in.

9. a 2.5 km., b=4 metres.

10. a mile, b=1 inch.

=

11. The area of a rectangle is 30 sq. cm., and its length is 6 cm. Find the breadth. Draw the rectangle on squared paper; and verify your work by counting the squares.

12. Find the length of a rectangle whose area is 3.9 sq. in., and breadth 1.5". Draw the rectangle on squared paper; and verify your work by counting the squares.

13. (i) When you treble the length of a rectangle without altering its breadth, how many times do you multiply the area?

(ii) When you treble both length and breadth, how many times do you multiply the area?

Draw a figure to illustrate your answers; and state a general rule.

14. In a plan of a rectangular garden the length and breadth are 3'6" and 2.5", one inch standing for 10 yards. Find the area of the garden.

If the area is increased by 300 sq. yds., the breadth remaining the same, what will the new length be? And how many inches will represent it on your plan?

15. Find the area of a rectangular enclosure of which a plan (scale 1 cm. to 20 metres) measures 6.5 cm. by 4.5 cm.

16. The area of a rectangle is 1440 sq. yds. If in a plan the sides of the rectangle are 3.2 cm. and 4.5 cm., on what scale is the plan drawn?

17. The area of a rectangular field is 52000 sq. ft. On a plan of this, drawn to the scale of 1" to 100 ft., the length is 3.25". What is the breadth?

Calculate the areas of the enclosures of which plans are given below. All the angles are right angles, and the dimensions are marked in feet.

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Calculate the areas represented by the shaded parts of the following plans. The dimensions are marked in feet.

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