1. RECTANGLE
![Picture](/uploads/2/2/9/3/22930296/896667.gif?493)
AREA OF A RECTANGLE
In coordinate geometry, the area of a rectangle is calculated in the usual way once the width and height are found. See Rectangle definition (coordinate geometry) to see how the width and height are found. Once the width and height are known the area is found by multiplying the width by the height in the usual way. The formula for the area is:
Area = width x height
A= W X H
PERIMETER OF A RECTANGLE
Perimeter of a Rectangle Recall that: The perimeter is the distance around a closed plane figure.The perimeter, P, of rectangle is given by the formula
P = 2(l + w)
where l is the length and w is the width of the rectangle.
EXAMPLE1:
Find the perimeter of a rectangular field of length 65 m and width 50 m.
Solution:
P= 2(L + W)
P= 2(65 M + 50 M)
P= 2(115 M)
P=230 METERS
So, the perimeter is 230 m.
EXAMPLE2:
Find the area of a rectangle whose length is 45 m and width is 23 m.
Solution:
A= L X W
A= 45M X 23M
A= 1035 SQUARE METERS
So, the area is 1035 square meters.
en.Wikipedia.org/wiki/Rectangle
In coordinate geometry, the area of a rectangle is calculated in the usual way once the width and height are found. See Rectangle definition (coordinate geometry) to see how the width and height are found. Once the width and height are known the area is found by multiplying the width by the height in the usual way. The formula for the area is:
Area = width x height
A= W X H
PERIMETER OF A RECTANGLE
Perimeter of a Rectangle Recall that: The perimeter is the distance around a closed plane figure.The perimeter, P, of rectangle is given by the formula
P = 2(l + w)
where l is the length and w is the width of the rectangle.
EXAMPLE1:
Find the perimeter of a rectangular field of length 65 m and width 50 m.
Solution:
P= 2(L + W)
P= 2(65 M + 50 M)
P= 2(115 M)
P=230 METERS
So, the perimeter is 230 m.
EXAMPLE2:
Find the area of a rectangle whose length is 45 m and width is 23 m.
Solution:
A= L X W
A= 45M X 23M
A= 1035 SQUARE METERS
So, the area is 1035 square meters.
en.Wikipedia.org/wiki/Rectangle
2. SQUARE
![Picture](/uploads/2/2/9/3/22930296/1305871.jpg?548)
Area The area of a square is calculated in the usual way once the length of a side is found. See Square definition (coordinate geometry) to see how the side length is found. Once the side length is known the area is found by multiplying the side length by itself in the usual way. The formula for the area is:
area = s2 where s is the length of any side (they are all the same). The "diagonals" method to find area If you know the length of a diagonal, the area is given by:
where
d is the length of either diagonal .The length of a diagonal can be found by using the the methods described in Distance between two points to find the distance between say A and C in the figure above.
Perimeter A square has four sides which are all the same length. The perimeter of a square (the total distance around the edge) is therefore the four times the length of any side. The formula for the perimeter is perimeter = 4s where s is the length of any side (they are all the same). Example The example below assumes you know how to calculate the side length of the square, as described in Square (Coordinate Geometry).
The perimeter of a square: To find the perimeter of a square, just add up all the lengths of the sides:
The area of a square: To find the area of a square, multiply the lengths of two sides together... Another way to say this is to say "square the length of a side!"
The sides and angles of a square:
The sides of a square are all congruent (the same length.)
The angles of a square are all congruent (the same size and measure.)
Remember that a 90 degree angle is called a "right angle." So, a square has four right angles.
Opposite angles of a square are congruent.
Opposite sides of a square are congruent.
Opposite sides of a square are parallel.
The diagonal of a square:
To find the length of the diagonal of a square, multiply the length of one side by the square root of 2: If the length of one side is x...
length of diagonal = x
The central angle of a square:
The diagonals of a square intersect (cross) in a 90 degree angle. This means that the diagonals of a square are perpendicular.The diagonals of a square are the same length (congruent).
Square Example :
Case 1: Find the area, perimeter and diagonal of a square with the given side 3.
Step 1: Find the area.
Area = (a)² = (3)² = 9.
Step 2: Find the perimeter.
Perimeter = 4(a) = 4 * 3 = 12.
Step 3: Find the diagonal.
Diagonal = (a)*(square root(2)) = 3 * 1.41 = 4.23.
www.mathsisfun.com/area.html
area = s2 where s is the length of any side (they are all the same). The "diagonals" method to find area If you know the length of a diagonal, the area is given by:
where
d is the length of either diagonal .The length of a diagonal can be found by using the the methods described in Distance between two points to find the distance between say A and C in the figure above.
Perimeter A square has four sides which are all the same length. The perimeter of a square (the total distance around the edge) is therefore the four times the length of any side. The formula for the perimeter is perimeter = 4s where s is the length of any side (they are all the same). Example The example below assumes you know how to calculate the side length of the square, as described in Square (Coordinate Geometry).
The perimeter of a square: To find the perimeter of a square, just add up all the lengths of the sides:
The area of a square: To find the area of a square, multiply the lengths of two sides together... Another way to say this is to say "square the length of a side!"
The sides and angles of a square:
The sides of a square are all congruent (the same length.)
The angles of a square are all congruent (the same size and measure.)
Remember that a 90 degree angle is called a "right angle." So, a square has four right angles.
Opposite angles of a square are congruent.
Opposite sides of a square are congruent.
Opposite sides of a square are parallel.
The diagonal of a square:
To find the length of the diagonal of a square, multiply the length of one side by the square root of 2: If the length of one side is x...
length of diagonal = x
The central angle of a square:
The diagonals of a square intersect (cross) in a 90 degree angle. This means that the diagonals of a square are perpendicular.The diagonals of a square are the same length (congruent).
Square Example :
Case 1: Find the area, perimeter and diagonal of a square with the given side 3.
Step 1: Find the area.
Area = (a)² = (3)² = 9.
Step 2: Find the perimeter.
Perimeter = 4(a) = 4 * 3 = 12.
Step 3: Find the diagonal.
Diagonal = (a)*(square root(2)) = 3 * 1.41 = 4.23.
www.mathsisfun.com/area.html
3. TRAPEZOID
![Picture](/uploads/2/2/9/3/22930296/89152.jpg?416)
The perimeter of a trapezoid:
To find the perimeter of a trapezoid, just add up all the lengths of the sides:
Perimeter = a + b + c + B
The area of a trapezoid:
To find the area of a trapezoid... The longer base (the bottom) is big B and the smaller base (the top) is little b.
But, this is double of what we need... So, multiply by 1/2!
Area=1/2(B+b)h
The sides and angles of a trapezoid:
The bases (top and bottom) of a trapezoid are parallel.
That's it. No sides needs to be congruent and no angles need to be congruent. Nothing special happens with the diagonals.
A special trapezoid is the isosceles trapezoid (like an isosceles triangle)...
Properties of the sides of an isosceles trapezoid:
The bases (top and bottom) of an isosceles trapezoid are parallel.
Opposite sides of an isosceles trapezoid are the same length (congruent).
The angles on either side of the bases are the same size/measure (congruent).
The diagonals (not show here) are congruent.
Adjacent angles (next to each other) along the sides are supplementary. This means that their measures add up to 180 degrees.
http://www.coolmath.com/reference/trapezoids.html
To find the perimeter of a trapezoid, just add up all the lengths of the sides:
Perimeter = a + b + c + B
The area of a trapezoid:
To find the area of a trapezoid... The longer base (the bottom) is big B and the smaller base (the top) is little b.
But, this is double of what we need... So, multiply by 1/2!
Area=1/2(B+b)h
The sides and angles of a trapezoid:
The bases (top and bottom) of a trapezoid are parallel.
That's it. No sides needs to be congruent and no angles need to be congruent. Nothing special happens with the diagonals.
A special trapezoid is the isosceles trapezoid (like an isosceles triangle)...
Properties of the sides of an isosceles trapezoid:
The bases (top and bottom) of an isosceles trapezoid are parallel.
Opposite sides of an isosceles trapezoid are the same length (congruent).
The angles on either side of the bases are the same size/measure (congruent).
The diagonals (not show here) are congruent.
Adjacent angles (next to each other) along the sides are supplementary. This means that their measures add up to 180 degrees.
http://www.coolmath.com/reference/trapezoids.html
4. RHOMBUS
![Picture](/uploads/2/2/9/3/22930296/244769.jpg?462)
The perimeter of a rhombus:
To find the perimeter of a rhombus, just add up all the lengths of the sides:
P= X+X+X+X
P= 4X
The area of a rhombus:
To find the area of a rhombus, multiply the lengths of the two diagonals and divide by 2 (same as multiplying by 1/2):
A= 1/2(ab)
The sides and angles of a rhombus:
The sides of a rhombus are all congruent (the same length.)
Opposite angles of a rhombus are congruent (the same size and measure.)
Properties of the diagonals of a rhombus:
The intersection of the diagonals of a rhombus form 90 degree (right) angles. This means that they are perpendicular. The diagonals of a rhombus bisect each other. This means that they cut each other in half.
Properties of the angles of a rhombus:
Adjacent sides (ones next to each other) of a rhombus are supplementary. This means that their measures add up to 180 degrees.
www.webcrawler.com
To find the perimeter of a rhombus, just add up all the lengths of the sides:
P= X+X+X+X
P= 4X
The area of a rhombus:
To find the area of a rhombus, multiply the lengths of the two diagonals and divide by 2 (same as multiplying by 1/2):
A= 1/2(ab)
The sides and angles of a rhombus:
The sides of a rhombus are all congruent (the same length.)
Opposite angles of a rhombus are congruent (the same size and measure.)
Properties of the diagonals of a rhombus:
The intersection of the diagonals of a rhombus form 90 degree (right) angles. This means that they are perpendicular. The diagonals of a rhombus bisect each other. This means that they cut each other in half.
Properties of the angles of a rhombus:
Adjacent sides (ones next to each other) of a rhombus are supplementary. This means that their measures add up to 180 degrees.
www.webcrawler.com
5. CIRCLE
![Picture](/uploads/2/2/9/3/22930296/6734702.png?450)
CIRCUMFERENCE OF A CIRCLE
Further information: Pi The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by:
C= 2πr
AREA OF A CIRCLE
Area enclosed by a circle = π × area of the shaded square Main article: Area of a disk As proved by Archimedes, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius which comes to π multiplied by the radius squared:
A= πr2
Equivalently, denoting diameter by d,
that is, approximately 79 percent of the circumscribing square (whose side is of length d).
The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.
Equations Cartesian coordinates Circle of radius r = 1, center (a, b) = (1.2, −0.5) In an x–y Cartesian coordinate system, the circle with center coordinates (a, b) and radius r is the set of all points (x, y) such that
This equation, also known as Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x − a and y − b. If the circle is centered at the origin (0, 0), then the equation simplifies to
The equation can be written in parametric form using the trigonometric functions sine and cosine as
where t is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the x-axis. An alternative parametrization of the circle is:
In this parametrization, the ratio of t to r can be interpreted geometrically as the stereographic projection of the circle onto the line passing through the center parallel to the x-axis.
In homogeneous coordinates each conic section with equation of a circle is of the form
It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points I(1: i: 0) and J(1: −i: 0). These points are called the circular points at infinity.
Polar coordinates In polar coordinates the equation of a circle is:
where a is the radius of the circle, is the polar coordinate of a generic point on the circle, and is the polar coordinate of the centre of the circle (i.e., r0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x-axis to the line connecting the origin to the centre of the circle). For a circle centred at the origin, i.e. r0 = 0, this reduces to simply r = a. When r0 = a, or when the origin lies on the circle, the equation becomes
In the general case, the equation can be solved for r, giving
the solution with a minus sign in front of the square root giving the same curve.
Complex plane In the complex plane, a circle with a centre at c and radius (r) has the equation . In parametric form this can be written .
The slightly generalised equation for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with , since . Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line.
Tangent lines Main article: Tangent lines to circles The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x1, y1) and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x1, y1), so it has the form (x1 − a)x + (y1 – b)y = c. Evaluating at (x1, y1) determines the value of c and the result is that the equation of the tangent is
or
If y1 ≠ b then the slope of this line is
This can also be found using implicit differentiation.
www.webcrawler.com
Further information: Pi The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by:
C= 2πr
AREA OF A CIRCLE
Area enclosed by a circle = π × area of the shaded square Main article: Area of a disk As proved by Archimedes, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius which comes to π multiplied by the radius squared:
A= πr2
Equivalently, denoting diameter by d,
that is, approximately 79 percent of the circumscribing square (whose side is of length d).
The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.
Equations Cartesian coordinates Circle of radius r = 1, center (a, b) = (1.2, −0.5) In an x–y Cartesian coordinate system, the circle with center coordinates (a, b) and radius r is the set of all points (x, y) such that
This equation, also known as Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x − a and y − b. If the circle is centered at the origin (0, 0), then the equation simplifies to
The equation can be written in parametric form using the trigonometric functions sine and cosine as
where t is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the x-axis. An alternative parametrization of the circle is:
In this parametrization, the ratio of t to r can be interpreted geometrically as the stereographic projection of the circle onto the line passing through the center parallel to the x-axis.
In homogeneous coordinates each conic section with equation of a circle is of the form
It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points I(1: i: 0) and J(1: −i: 0). These points are called the circular points at infinity.
Polar coordinates In polar coordinates the equation of a circle is:
where a is the radius of the circle, is the polar coordinate of a generic point on the circle, and is the polar coordinate of the centre of the circle (i.e., r0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x-axis to the line connecting the origin to the centre of the circle). For a circle centred at the origin, i.e. r0 = 0, this reduces to simply r = a. When r0 = a, or when the origin lies on the circle, the equation becomes
In the general case, the equation can be solved for r, giving
the solution with a minus sign in front of the square root giving the same curve.
Complex plane In the complex plane, a circle with a centre at c and radius (r) has the equation . In parametric form this can be written .
The slightly generalised equation for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with , since . Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line.
Tangent lines Main article: Tangent lines to circles The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x1, y1) and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x1, y1), so it has the form (x1 − a)x + (y1 – b)y = c. Evaluating at (x1, y1) determines the value of c and the result is that the equation of the tangent is
or
If y1 ≠ b then the slope of this line is
This can also be found using implicit differentiation.
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6. PENTAGON
![Picture](/uploads/2/2/9/3/22930296/3537924.gif?404)
AREA OF A PENTAGON
Area of a Pentagon is the amount of space occupied by the pentagon. Pentagon is defined as a polygon which has 5 sides that are equal. The 5 angles present in the pentagon are also equal. A pentagon can be sectored into 5 similar triangles. The measurement of each interior angle in a regular pentagon 108 degrees.
A= 5/2sa
EXAMPLE
Question 1: Find the area of a pentagon of side 5 cm and apothem length 3 cm?
Solution:
Given,
s = 5 cm
a = 3 cm
Area of a pentagon
= 5/2sa
= 5/2 * 5 * 3 cm2
= 5∗5∗32 cm2
= 752 cm2
= 37.5 cm2
Area of a Pentagon is the amount of space occupied by the pentagon. Pentagon is defined as a polygon which has 5 sides that are equal. The 5 angles present in the pentagon are also equal. A pentagon can be sectored into 5 similar triangles. The measurement of each interior angle in a regular pentagon 108 degrees.
A= 5/2sa
EXAMPLE
Question 1: Find the area of a pentagon of side 5 cm and apothem length 3 cm?
Solution:
Given,
s = 5 cm
a = 3 cm
Area of a pentagon
= 5/2sa
= 5/2 * 5 * 3 cm2
= 5∗5∗32 cm2
= 752 cm2
= 37.5 cm2
PERIMETER OF A PENTAGON
![Picture](/uploads/2/2/9/3/22930296/4446890.jpg?775)
Perimeter of a regular pentagon is the sum of its sides.
www.mathsisfun.com/geometry/pentagon.html