Area of plane figures

The idea of area may be explained as the amount of space enclosed within the boundary of a figure. For instance, the area of the floor of a classroom is the amount of space enclosed within the four corners of the room. Also, the area of the top of the teacher’s table is the amount of space enclosed within the edges of the table.

To measure this amount of space, we determine the number of square units. As an example, let us find the area of the rectangular floor of length 10cm and width 6cm. To do this, one method may be to take a square cardboard of one centimeter side and starting from one corner of the floor, mark the outline of the cardboard, edge to edge, until the whole space of the floor is covered. Then the number of the one-square centimeter marking is counted and that gives the area of the floor in square centimeters.

(A) To construct a triangle equal area to give a given triangle

(a) When the triangles are on equal bases

1. Draw the given triangle ABC and produce the base AB to D, making DE = AB.
2. Through C, draw CF parallel to AD.

iii. With centre E and radius equal to a side of the required triangle, cut CF at G.

(B) To construct a triangle given its base and its area

Suppose that the given base and area are 5cm and 9sq. cm respectively:

1. Divide the given area by the given base, i.e. 9/5 =1.8cm.
2. Draw a rectangle ABCD whose length is 5cm and width is 1.8cm.

iii. Produce the width of the rectangle to twice it magnitude, i.e. mark off CE = BC = 1.8cm.

1. Join E. Triangle ABE is the required triangle.

(C) To construct a triangle equal in area to any to any given parallelogram

1. Draw the given parallelogram ABCD and draw diagonal BD.
2. Through C, draw a line parallel to DB to intersect AB produced at E.

iii. Join DE. Triangle AED is the required triangle.

(D) To construct a rectangle equal in area to a given rectangle of different length

1. Draw the given rectangle ABCD.
2. On AB (produced), mark off AE equal to the different length of the required rectangle.

iii. Join DE.

1. Through B, draw a line parallel to ED to intersect AD (produced)at F. AF is the width of the required rectangle AEGF.

(E) To construct a square in area to a given rectangle

1. Draw the given rectangle ABCD.

With centre B and radius BC, swing arc CE to intersect AB produced at E.

iii. Bisect AE in F and draw a semi-circle on AE diameter.

1. Produce BC to meet the semi-circle at G. BG is the side of the required square.
2. Complete square BGHI.

(F) To construct a square equal in area to the sum of the area of two given squares

1. Draw a line and mark off AB equal to the side of one of the given squares.
2. At A, erect a perpendicular and mark off AC equal to the side of the other given square.

iii. Join CB. CB is the side of the required square.

1. Complete the required square CBDE.

Note: This is based on Pythagoras’ theorem. This theorem states that the square on the hypothenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.

Enlargement and Reduction of Regular Plane Figures

(A) To construct a figure similar to a given figure ABCDEF with its sides in the ratio of 6.4 to those of the given figure.

1. Draw the given figure ABCDE.
2. Divide the figures into triangle by drawing lines AC, AD and AE.

iii. Draw line AG at a convenient angle and set off on it from A, 6 equal parts.

1. Join point 4 to B and draw GB1 parallel to GB to cut AB produced at B1.
2. Draw B1, C1, D1, and D1 E1 parallel to BC, CD, and DE respectively to complete the required figure.

(B) To construct a figure similar to a given figure ABCDEF with its sides in the ratio of 4:6 to those of the given figure

1. Draw the given figure ABCDEF.
2. Divide the figures into triangles by drawing lines AC, AD and AE.

iii. Divide AB into six equal parts.

1. Draw B1, C1, C1, D1 and D1 E1 parallel to BD, CD and DE respectively to complete the required figure.

(C) To construct the size of a given rectangle by a given proportion

Let the proportion be 6:4.

1. Draw the given rectangle ABCD.
2. Choose point P at any convenient distance from the rectangle, and from radiate lines to corner A, B, C and D.

iii. Divide PA into 6 equal parts.

1. Draw A1, D1, D1, C1 and B1 A1 respectively to complete the required rectangle.

 ASSESSMENT

1. Define Area?