AREA & VOLUME

Instructors :…
Presentation group 4: …


Area of a rectangle
Rectangle: A 4-sided polygon where all interior angles are 90°

The area of a rectangle is given by multiplying the width times the height. As a
formula:
Area=wh

•

Where
w  is the width
h  is the height

Rectangle: Hình chữ nhật
Interior angle: góc phía trong

Width: Chiều rộng
Height: Chiều dài

Side: cạnh


Area of a Square
Square: A 4-sided regular polygon with all sides equal and all internal angles 90°

 

The area of a square is given by the formula
area=width×height
But since the width and height are by definition the same, the formula is usually
written as

where s is the length of one side.

Square: hình vng

Regular polygon: đa giác đều

Length: chiều dài


Triangle
01

Triangle: A closed figure consisting of three
line segments linked end-to-end.

04

Right Triangle: A triangle where
one of its interior angles is a 
right angle (90 degrees).

A 3-sided polygon.


Isosceles Triangle:

02

Acute triangle: A triangle where all

A triangle which has two of its sides

05

equal in length.

three internal angles are acute (less
than 90 degrees).

03

Equilateral Triangle: A triangle which

Obtuse Triangle

has all three of its sides equal in length.

06

A triangle where one of the internal
angles is obtuse (greater than 90
degrees).

A closed figure: hình khép kín


segment: đoạn

linked end-to-end: liên kết từ đầu đến cuối


Area of a triangle
●

 

The area of a triangle is given by the formula below.

Where




a  is the length of the base
h  is the length of the corresponding altitude.

Triangle: Tam giác

Base: Đáy

Corresponding: Tương ứng

Altitude: Chiều cao



Heron's Formula for the area of a triangle (Hero's Formula)

●  
A method for calculating the area of a triangle when you
know the lengths of all three sides.

●

Let a,b,c be the lengths of the sides of a triangle. The
area is given by:

where p is half the perimeter, or 

Half the perimeter: Nửa chu vi


Area of an equilateral triangle

●  
The area of an equilateral triangle (all sides congruent)
can be found using the formula

where s is the length of one side of the triangle.

Equilateral triangle: tam giác đều

Congruent: Đồng/ đều/ cùng kích thước


Area of a parallelogram

Parallelogram: A quadrilateral with both pairs of opposite sides parallel.

The area of a parallelogram is given by the formula
Area=b.a

•

Where
b is the length of any base
a is the corresponding altitude

Parallel: song song
Parallelogram: Hình bình hành


Area of a rhombus
Rhombus : A quadrilateral with all four sides equal in length.

1. The "base times height" method
The area of a rhombus is given by the formula:
area=b.a
where
b is the length of the base
a is the altitude (height).

Perpendicular: vng góc
Distance: khoảng cách


Area of a rhombus


 

2. The "diagonals" method
Another simple formula for the area of a rhombus when you know the lengths of the diagonals. The area is half the
product of the diagonals. As a formula:

where
d1 is the length of a diagonal
d2 is the length of the other diagonal

Diagonal: đường chéo


Area of a rhombus

 

3. Using trigonometry
If you are familiar with trigonometry, there is a handy formula when you know the length of a side and any angle:

where
s is the length of any side
a is any interior angle
sin  is the sine function

Trigonometry: lượng giác

Handy: hữu ích



Area of an inscribed (cyclic) quadrilateral.
Brahmagupta's Formula
Inscribed (cyclic) quadrilateral: A quadrilateral where all four vertices lie on a common circle.

 

A formula for calculating the area of an inscribed, or cyclic quadrilateral
when you know the lengths (a,b,c,d) of the sides.

Where a,b,c,d are side lengths, and p is half the perimeter:

Cyclic/ Inscribed : Nội tiếp

Inscribed quadrilateral: Tứ giác nội tiếp


Area of a trapezoid
Trapezoid: A quadrilateral which has at least one pair of parallel sides

 

The area of a trapezoid is basically the average width times the altitude, or as a
formula: 

where
are the lengths of each base
h is the altitude (height)

Trapezoid: Hình thang


Basically: Về cơ bản

The average: Trung bình


Area of a regular polygon
Regular Polygon: A polygon that has all sides equal and all interior angles equal

 1.

Given the length of a side.

By definition, all sides of a regular polygon are equal in length. If you know the length of one of the
sides, the area is given by the formula: 

Where
a  is the length of any side
n  is the number of sides
tan  is the tangent function calculated in degrees

Regular polygon: đa giác đều
Degrees: Độ
Tangent function: Hàm tiếp tuyến


Area of a regular polygon

 


2. Given the radius (circumradius)
If you know the radius (distance from the center to a vertex):



Where
r  is the radius (circumradius)
n  is the number of sides
sin  is the sine function calculated in degrees


Area of a regular polygon
 

3. Given the apothem (inradius)
If you know the apothem, or inradius, (the perpendicular distance from center
to a side.), the area is given by: 

where
a is the length of the apothem (inradius)
n  is the number of sides
tan  is the tangent function calculated in degrees

Apothem: A line segment from the center of a 
regular polygon to the midpoint of a side.
The apothem is also the radius of the 
incircle of the polygon


of a

 

Where:
r: Radius of a circle
= 3,14

 

circle 
If you know the diameter

The radius r of a circle is half the diameter d

Substituting r into the area formula


VOLUME


Volume enclosed by a cube

Cube :A solid with six congruent square faces. A regular hexahedron.

 

The volume of a cube have a formula:

where: s  is the length of any edge of the cube.

Cube: Khối lập phương


Volume: Thể tích

Edge: Cạnh/ cạnh biên


Volume enclosed by a cylinder

Cylinder: A cylinder is a closed solid that has two
parallel (usually circular) bases connected by a curved
surface.

 The

volume is found by multiplying the area of one end of the cylinder (base) by its

height.
Since the end (base) of a cylinder is a circle, the area of that circle is given by the
formula:

Multiplying by the height h we get

where:
π  is Pi, approximately 3.142
r  is the radius of the circular end of the cylinder
h  height of the cylinder
Cylinder: Hình trụ
Circular : vịng quanh

Approximately: Xấp xỉ

Curved surface: mặt cong


Volume of a sphere

 

The volume enclosed by a sphere is given by the
formula

Where r is the radius of the sphere.

Sphere: Hình cầu


Volume of a cone
Cone: A solid that has a circular base and a single vertex.

 

The volume enclosed by a cone is given by the
formula

Where

•

 r is the radius of the circular base of the cone

•


h is its height.

Cone: Hình nón
Radius: bán kính


Volume of a pyramid
Pyramid: A polyhedron that has a base and three or more triangular faces that meet at a point above the base (the
apex).

 The

volume enclosed by a pyramid is one third of the base area times the perpendicular height. As a
formula:

Where:
b is the area of the base of the pyramid
h is its height. The height must be measured as the vertical distance from the apex down to the
base.

Pyramid: Khối chóp

Apex: Đỉnh

Vertical: thẳng đứng


Volume of a triangular prism


A prism has two congruent, parallel faces called the bases of the prism. The volume of any
prism can be found by multiplying the area of one of the bases by its height. In the case of a
triangular prism, each base is a triangle. As a formula
V=ah
where:
a is the area of one triangular end face.
h is the height.

Prism: Lăng trụ
Triangular prism: Lăng trụ tam giác
Congruent: đồng dạng


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