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Geometry

Perimeter of Polygons

The perimeter of a polygon is the length of the distance around the outside of the polygon. To find the perimeter of a polygon, take the sum of the distance along each of its sides.

Although a polygon is two-dimensional, perimeter is one-dimensional and is measured in linear units. Think of a fence around a rectangular back yard. The yard itself is two-dimensional: it has a length and a width. The amount of fence needed to enclose the yard (the perimeter) is one-dimensional (length). If you went to a lumber store to buy fencing, you would need to ask for a length of fence in feet, yards, or meters. The shapes below are much smaller than a fence. Therefore, smaller units, such as centimeters and inches, are used.

Example 1:	A rectangle has a length of 8 cm and a width of 3 cm. Find its perimeter.
Solution:	P = 8 cm + 8cm + 3 cm + 3 cm = 22 cm

Example 2:	Find the perimeter of a square the sides of which are 2 in long.
Solution:	P = 2 in + 2 in + 2 in + 2 in = 8 in

Example 3:	Find the perimeter of an equilateral triangle the sides of which are 4 cm long.
Solution:	P = 4 cm + 4 cm + 4 cm = 12 cm

A square and an equilateral triangle are examples of a regular polygon. Another way to find the perimeter of a regular polygon is to multiply the number of sides by the distance along one side.

Example 4:	Find the perimeter of a regular pentagon the sides of which are 3 in long.
Solution:	P = 5 · (3 in) = 15 in

Area of Squares and Rectangles

The area of a polygon is the number of square units inside that polygon. Area is 2-dimensional like a room carpet or an area rug.

To find the area of a square, multiply the distance along the side of the square by itself. The formula is:

A = s · s

where s represents the distance along the side of the square.

To find the area of a rectangle, multiply its length by its width. The formula is:

A = (L) · (W)

where L is the length and W is the width of the rectangle.

Example 1:	Find the area of a square the sides of which are 2 in long.
Solution:	A = s · s
	A = (2 in) · (2 in) = 4 in²

Example 2:	A rectangle has a length of 8 cm and a width of 3 cm. Find its area.
Solution:	A = (L) · (W)
	A = (8 cm) · (3 cm) = 24 cm²

Example 3:	The area of a rectangle is 12 square inches and its width is 3 in. What is its length?
Solution:	A = (L) · (W)
	12 = (L) · (3)
	L = 4 in

Area of Parallelograms

The area of a polygon is the number of square units inside that polygon. Area is 2-dimensional like a room carpet or an area rug.

A parallelogram is a 4-sided shape formed by two pairs of parallel lines. Opposite sides are equal in length and opposite angles are equal in size. To find the area of a parallelogram, multiply its base by its height. The formula is:


A =	(b · h)	where b is the base and h is the height.

The base and height of a parallelogram must be perpendicular. The lateral sides of a parallelogram are not perpendicular to the base, so a dotted line is drawn to represent the height.

Example 1:	Find the area of a parallelogram with a base of 12 cm and a height of 5 cm.
Solution:	A = (b) · (h)
	A = (12 cm) · (5 cm)
	A = 60 cm²

Example 2:	Find the area of a parallelogram with a base of 7 in and a height of 10 in.
Solution:	A = (b) · (h)
	A = (7 in) · (10 in)
	A = 70 in²

Example 3:	The area of a parallelogram is 24 square centimeters and its base is 4 cm. What is its height?
Solution:	A = (b) · (h)
	24 = (4) · (h)
	h = 6 cm

Types of Triangles

Triangles are classified into various types, using two different parameters - the lengths of their sides and the measure of their angles. Length of the Side Based on the lengths of their sides, triangles are classified into three categories.
Equilateral triangle : If the lengths of all three sides of the triangle are equal, then it is called an equilateral triangle.
Isosceles triangle : If only two sides of a triangle are equal in length, it is called as an isosceles triangle.
Scalene triangle : If all the sides of a triangle have different lengths it is called a scalene triangle.
Angles
Acute triangle : A triangle in which all the angles are acute, ( i.e. < 90⁰ ) is called as an acute triangle.
A special case of an acute triangle is when all the three acute angles are equal. This D is called an equiangular triangle. Since the sum of all the angles of a triangle is 180⁰, it can be said that each angle of an equiangular triangle is 60⁰ .
Obtuse triangle : A triangle in which one of the angles is obtuse is called as an obtuse triangle. Since the sum of all the angles of a triangle is 180⁰ it can be said that the other two angles of an obtuse triangle are acute.
Right Triangle : It is a triangle in which one of the angles is a right angle. Since Ð KJL is 90⁰ it can be said that Ð JKJL and Ð JLK are complementary. In a right triangle the side opposite to the right angle is called the hypotenuse.

Area of Triangles

The area of a polygon is the number of square units inside that polygon. Area is 2-dimensional like a room carpet or an area rug.

To find the area of a triangle, multiply its base by its height, then divide by 2. The division by 2 comes from the fact that a parallelogram can be divided into 2 triangles. For example, in the diagram to the right, the area of each triangle is equal to one-half the area of the parallelogram. The formula for the area of a triangle is:


A =	¹/₂ · (b · h)	where b is the base and h is the height.

The base and height of a triangle must be perpendicular. In each of the triangles below, the base is a side of the triangle. However, the height may or may not be a side of the triangle. For example, in the right triangle in Example 2, the height is a side of the triangle since it is perpendicular to the base. In other types of triangles, the lateral sides are not perpendicular to the base, so a dotted line is drawn to represent the height (see Examples 1 & 3).

Example 1:	Find the area of an acute triangle with a base of 15 in and a height of 4 in.
Solution:	A = ¹/₂ · (15 in) · (4 in)
	A = ¹/₂ · (60 in²)
	A = 30 in²

Example 2:	Find the area of a right triangle with a base of 6 cm and a height of 9 cm.
Solution:	A = ¹/₂ · (6 cm) · (9 cm)
	A = ¹/₂ · (54 cm²)
	A = 27 cm²

Example 3:	Find the area of an obtuse triangle with a base of 5 in and a height of 8 in.
Solution:	A = ¹/₂ · (5 in) · (8 in)
	A = ¹/₂ · (40 in²)
	A = 20 in²

Circumference of a Circle

A circle is a shape with all points the same distance from its center. It is named by its center. The circle to the left is called circle A since its center is at point A.

If you measure the distance around a circle and divide it by the distance across the circle through its center, you will always come close to a particular value, depending upon the accuracy of your measurement. This value is approximately 3.14159265358979323846... We use the Greek letter

(Pi) to represent this value. Using computers, mathematicians have been able to calculate the value of

to thousands of places.

The distance around a circle is called its circumference. The distance across a circle through its center is called its diameter.

is the ratio of the circumference of a circle to its diameter. For any circle, if you divide its circumference by its diameter, you get a value close to

. This relationship is expressed in the following formula: ^C/_D =

where C is the circumference and D is the diameter. You can test this formula at home with a dinner plate. If you measure the circumference and the diameter of the plate and then divide the circumference by the diameter, your quotient should come close to

. Another way to write this formula is:

C = · D

The radius of a circle is the distance from the center of a circle to a point on the circle. If you place two radii end-to-end in a circle, you would have the same length as one diameter. So a circle's diameter is twice as long as its radius. This relationship is expressed in the following formula: D = 2 · R where D is the diameter and R is the radius.

Circumference, diameter and radius are measured in linear units, such as inches and centimeters. A circle has many different radii and many different diameters, each passing through its center. A real-life example of a radius is the spoke of a bicycle wheel. A 9-inch pizza is an example of a diameter.

Example 1:	The diameter of a circle is 3 cm. What is its circumference? (Use = 3.14)
Solution:	C = · D
	C = 3.14 · (3 cm)
	C = 9.42 cm

Example 2:	The radius of a circle is 2 in. What is its circumference? (Use = 3.14)
Solution:	D = 2 · R
	D = 2 · (2 in)
	D = 4 in
	C = · D
	C = 3.14 · (4 in)
	C = 12.56 in

Example 3:	The circumference of a circle is 15.7 cm. What is its diameter? (Use = 3.14)
Solution:	C = · D
	15.7 cm = 3.14 · D
	D = 15.7 cm ÷ 3.14
	D = 5 cm

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