Geometry: Quadrilaterals

In these lessons, we will learn

• quadrilaterals.
• types of quadrilaterals: parallelogram, rhombus, rectangle, square, kite,
trapezoid (trapezium).
What Is A Quadrilateral?
Quadrilaterals are two-dimensional four-sided polygons on a plane.
Quadrilaterals have the following properties:
• four sides,
• two diagonals,
• four internal angles (or vertices),
• the interior angles add up to 360°.
The following diagram shows the properties of some quadrilaterals: square,
rectangle, parallelogram, trapezoid, rhombus, kite. Scroll down the page for
more examples on the properties of a quadrilaterals.

A convex quadrilateral has all interior angles less than 180°.

A concave quadrilateral has at least one interior angle greater than 180°.
(memory tool: concave has a "cave" in it)

Angles In A Quadrilateral Worksheet

Types Of Quadrilaterals
We will describe the following types of
quadrilaterals: parallelogram, rhombus, rectangle, square, trapezoid, kite and tr
apezium.
Parallelogram
A parallelogram is a four-sided polygon that has the following properties:
• opposite sides are parallel,
• opposite sides are equal,
• opposite angles are equal,
• diagonals bisect each other.

Rhombus
A rhombus is a four-sided polygon that has the following properties:
• opposite sides are parallel,
• all four sides are equal,
• opposite angles are equal,
• diagonals bisect each other at right angles,
• two lines of symmetry (which are the diagonals).
A rhombus is a special case of a parallelogram with four equal sides.

Rectangle
A rectangle is a four-sided polygon that has the following properties:
 • opposite sides are parallel,
• opposite sides are equal,
• its angles are all right angles (i.e. 90°),
• diagonals bisect each other,
• diagonals are equal,
• two lines of symmetry.
A rectangle is a special case of a parallelogram with all angles equal to 90°.

Square
A square is a four-sided polygon that has the following properties:
• opposite sides are parallel,
• all sides are equal,
• its angles are all right angles (i.e. 90°),
• diagonals bisect each other,
• diagonals are equal,
• four lines of symmetry.
A square is a special case of a rectangle in which all the sides are equal.
Take note that a diagonal of a square makes two 45°-45°-90° triangles with the
sides of the square. Therefore, when given the length of a side, you will be able
to work out the length of the diagonal. Or when given the length of the
diagonal, you will be able to figure out the length of the side.
Trapezoid
There are two possible definitions for trapezoids.
A trapezoid is a four-sided polygon that has exactly one pair of opposite
sides parallel. Using this definition, a parallelogram would not be considered a
trapezoid.
A trapezoid is a four-sided polygon that has at least one pair of opposite
sides parallel. Using this definition, a parallelogram is a special case of a
trapezoid.
A trapezoid is also called a trapezium (UK English).

p is parallel to q l is parallel to m
An isosceles trapezoid has the following properties:
• the two non-parallel sides are equal in length,
• the base angles are equal.
A right trapezoid is a trapezoid with two right angles.

Kite
A kite is a four-sided polygon that has the following properties:
• two pairs of adjacent sides equal,
• one pair of opposite angles equal,
• longer diagonal bisects the shorter diagonal at right angles,
• one line of symmetry.

Trapezium
A trapezium represents a different shape depending on whether you are in the
US or not.
In the US, a trapezium is a quadrilateral that has no parallel sides.
(Outside the US, a quadrilateral that has no parallel sides is called an irregular
quadrilateral.)
Outside the US, a trapezium is a quadrilateral that has a pair of parallel sides.

(In the US, this quadrilateral is called a trapezoid.)

Geometry: Polygons
What are Polygons?

Polygons are two-dimensional many-sided figures on a plane, with sides that

are line segments. Some examples are: triangles, quadrilaterals, pentagons (5-
sided) and hexagons (6-sided).

Pentagon (5-sided polygon)

A regular polygon is a polygon with equal sides and equal angles.

Regular polygon
How to find the sum of angles in a polygon?

Sum of Angles in a Triangle

The sum of angles in a triangle is 180°.
For the sum of angles of other polygons, we can either divide the polygons into
triangles or use a formula.
Dividing polygons into triangles
For the other polygons, we can figure out the sum of angles by dividing the
polygons into triangles. Any polygon can be separated into triangles by drawing
all the diagonals that can be drawn from one single vertex.
In the quadrilateral shown below, we can draw only one diagonal from vertex A
to vertex B. So, a quadrilateral can be separated into two triangles.
The sum of angles in a triangle is 180°. Since a quadrilateral is made up of two
triangles the sum of its angles would be 180° × 2 = 360°
The sum of angles in a quadrilateral is 360°
Formula for the sum of angles
We can also use a formula to find the sum of the interior angles of any polygon.
If n is the number of sides of the polygon then,
sum of angles = (n - 2)180°
Example 1:
Find the sum of the interior angles of a hexagon (6-sided polygon)
Solution:
Step 1: Write down the formula (n - 2)180°
Step 2: Plug in the values(6 - 2)180° = (4)180° = 720°
Answer: The sum of the interior angles of a hexagon (6-sided) is 720°.

Geometry: Area of Polygons

In these lessons, we will learn how to use formulas to find the area of polygons:
square, rectangle, parallelogram, triangle, equilateral triangle, rhombus, kite,
trapezoid and the area of any regular polygon.
Area Of Polygons - Formulas
The area of a polygon measures the size of the region enclosed by the polygon.
It is measured in units squared.
The following table gives the formulas for the area of polygons.
Area Of A Square
The area of a square is equal to the length of one side squared.
Area of a square = s2
Area Of A Rectangle
The area of a rectangle" is equal to the product of the length of its base and the
length of its height. Sometimes, the height is called the "altitude". We can also
call the longer side the "length" and the shorter side the "width"

Area of a rectangle = lw
Worksheet 1, Worksheet 2, Worksheet 3 to calculate the area and perimeter of
rectangles.
Area Of A Parallelogram

To get the area of a parallelogram, we first draw a perpendicular line segment

from one corner of the parallelogram to the opposite side. This is the height ( h)
of the parallelogram. The area of a parallelogram is equal to the product of its
length and height. (Remember to always use the perpendicular height.)
Area of a parallelogram = lh
You may notice that lh is also the area of a rectangle with
dimensions l and h. The diagram below will explain why. If we cut out the
triangle ABC and add it to the other side (triangle DEF), you will have a rectangle
with dimensions l and h that has the same area as the original parallelogram.

Worksheet to calculate the area of parallelograms.

Area Of A Triangle

To get the area of a triangle, we first choose one of the sides to be the base (b).
Then we draw a perpendicular line segment from a vertex of the triangle to the
base. This is the height (h) of the triangle. The area of a triangle is equal to half
the product of the base and the height.

There are also other formulas for the area of a triangle. More examples,
formulas and videos for area of triangles

Area Of A Rhombus
We can obtain the area of a rhombus, given the lengths of its diagonals.
If the lengths of the diagonals are a and b, then area of the rhombus is equal to
half the product of the diagonals.

Area of rhombus =

If you are given the length of one side (s) and the perpendicular height (h) from
one side to the vertex then the area of the rhombus is equal to the product of
the side and height.
The area of the rhombus is given by the formula:
Area of rhombus = sh
This formula for the area of a rhombus is similar to the area formula for a
parallelogram.
Area Of A Kite
The area of a kite uses the same formula as the area of a rhombus. The area of
a kite is equal to half the product of the diagonals.

Area of kite =

Area Of A Trapezoid

a is parallel to b
To get the area of a trapezoid, we sum the length of the parallel sides and
multiply that with half of the height. Remember that the height needs to be
perpendicular to the parallel sides.
Area Of Any Regular Polygon
A regular polygon is a polygon where all the sides are the same length and all
the angles are equal.

How to find the area of a regular polygon?

The apothem of a regular polygon is a line segment from the center of the
polygon to the midpoint of one of its sides. The area of any regular polygon is
equal to half of the product of the perimeter and the apothem.

Area of regular polygon = where p is the perimeter and a is the apothem.

Geometry: Perimeter of Polygons

In these lessons, we will learn the perimeter of the following polygons:

• square
• rectangle
• parallelogram
• triangle
• rhombus
• trapezoid
We will also learn how to solve word problems involving perimeter of polygons.

Perimeter of Polygons
The perimeter of a polygon is the sum of the lengths of its sides. It is the
distance around the outside of the polygon.
See also area of circles, circumference of circles.
Perimeter of a Square
Since the sides of a square are equal, the perimeter of a square is 4 times the
length of its side.
Perimeter of a square = s + s + s + s = 4s
where s is the length of one side
Perimeter of a Rectangle
A rectangle has four sides, with opposite sides being equal in length.

Perimeter of a rectangle = l + l + w + w = 2(l + w),

where l is the length and w is the width of the rectangle.

Perimeter of a Parallelogram
Similar to a rectangle, a parallelogram has four sides with opposite sides being
equal in length.

Perimeter of a parallelogram = l + l + w + w = 2(l + w)

where l is the length and w is the width of the parallelogram.
Worksheet to calculate the area and perimeter of parallelograms
Perimeter of a Triangle
Perimeter of a triangle = a + b + c
where a, b and c are the lengths of each side of the triangle.
Perimeter of a Rhombus
A rhombus has 4 equal sides, so the perimeter of a rhombus is 4 times the
length of its side.

Perimeter of a rhombus = s + s + s + s = 4s
where s is the length of one side
Perimeter of a Trapezoid

a is parallel to b
Perimeter of a trapezoid = a + b + c + d
where a, b, c and d are the lengths of each side.

Geometry: Circles

These lessons cover the various properties of circles, the parts of a circle and
the terms commonly associated with circles.
We shall learn about circles and the properties of circles.
• diameter,
• chord,
• radius,
• arc,
• semicircle, minor arc, major arc,
 • tangent,
• secant,
• circumference and formula,
• area and formula,
• sector and formula.
The following figures show the different parts of a circle: tangent, chord,
radius, diameter, arc, segment, sector. Scroll down the page for more examples
and explanations.

Circle
In geometry, a circle is a closed curve formed by a set of points on a plane.
plane that are the same distance from its center O. That distance is known as
the radius of the circle.

Diameter
The diameter of a circle is a line segment that passes through the center of the
circle and has its endpoints on the circle. All the diameters of the same circle
have the same length.

Chord
A chord is a line segment with both endpoints on the circle. The diameter is a
special chord that passes through the center of the circle. The diameter would
be the longest chord in the circle.

Radius
The radius of the circle is a line segment from the center of the circle to a point
on the circle.

In the above diagram, O is the center of the circle and and are radii of
the circle. The radii of a circle are all the same length. The radius is half the

length of the diameter.

Arc
An arc is a part of a circle.
In the diagram above, the part of the circle from B to C forms an arc. It is called
arc BC.
An arc can be measured in degrees. In the circle above, the measure of arc BC is
equal to its central angle ∠BOC, which is 45°.
Semicircle, Minor Arc and Major Arc
A semicircle is an arc that is half a circle. A minor arc is an arc that is smaller
than a semicircle. A major arc is an arc that is larger than a semicircle.
Tangent
A tangent to a circle is a line that touches a circle at only one point. A tangent
is perpendicular to the radius at the point of contact.

In the above diagram, the line containing the points B and C is a tangent to the
circle.
It touches the circle at point B and is perpendicular to the radius
is perpendicular to i.e.

Secant
A secant is a straight line that cuts the circle at two points. A chord is the
portion of a secant that lies in the circle.
Circumference
The circumference of a circle is the distance around a circle.
Calculating the circumference of a circle involves a constant called pi with the
symbol π. The value of π (pi) is approximately 3.14159265358979323846…
but usually rounding to 3.142 should be sufficient. (see a mnemonic for π)
The formula for the circumference of a circle is
C = πd (see a mnemonic for this formula)
or
C = 2πr
where C is the circumference, d is the diameter and r is the radius.
If you are given the diameter then use the formula C = πd
If you are given the radius then use the formula C = 2πr

Example 1: Find the circumference of the circle with a diameter of 8 inches.

Solution:
Step 1: Write down the formula: C = πd
Step 2: Plug in the value: C = 8π
Answer: The circumference of the circle is 8π ≈ 25.163 inches.
Example 2: Find the circumference of the circle with a radius of 5 inches.
Solution:
Step 1: Write down the formula: C = 2πr
Step 2: Plug in the value: C = 10π
Answer: The circumference of the circle is 10π ≈ 31.42 inches.
Area
The area of a circle is the region enclosed by the circle. It is given by the
formula:
A = πr2 (see a mnemonic for this formula)
where A is the area and r is the radius.
Since the formula is only given in terms of radius, remember to change from
diameter to radius if necessary.
Circles Worksheet to calculate the area of a circle.
Area and Circumference Worksheet to calculate the area and circumference of a
circle.
Circle Problems Worksheet to calculate problems that involve the radius,
diameter, circumference and area of circle.
Example 1: Find the area the circle with a diameter of 10 inches.
Solution:
Step 1: Write down the formula: A = πr2

Step 2: Change diameter to radius:

Step 3: Plug in the value: A = π52 = 25π

Answer: The area of the circle is 25π ≈ 78.55 square inches.
Example 2: Find the area the circle with a radius of 10 inches.
Solution:
Step 1: Write down the formula: A = πr2
Step 2: Plug in the value: A = π102 = 100π
Answer: The area of the circle is 100π ≈ 314.2 square inches.
Sector
A sector is like a "pizza slice" of the circle. It consists of a region bounded by
two radii and an arc lying between the radii. The area of a sector is a fraction of
the area of the circle.

The formula to calculate the area of a sector is

Solid Geometry
In these lessons, we will look at the geometric properties of 3D solids, such as
cubes, cuboids, prisms, cylinders, cones, pyramids and spheres.
What Is Solid Geometry?
Solid geometry is concerned with three-dimensional shapes. Some examples of
three-dimensional shapes are cubes, rectangular
solids, prisms, cylinders, spheres, cones and pyramids. We will look at the
volume formulas and surface area formulas of the solids. We will also discuss
some nets of solids.
The following figures show some examples of shapes in solid geometry. Scroll
down the page for more examples, explanations and worksheets for each
shape.

The following table gives the volume formulas and surface area formulas for
the following solid shapes: Cube, Rectangular Prism, Prism, Cylinder, Sphere,
Cone, and Pyramid.
Cubes
A cube is a three-dimensional figure with six equal square faces.

The figure above shows a cube. The dotted lines indicate edges hidden from
your view.
If s is the length of one of its sides, then the volume of the cube is s × s × s
Volume of the cube = s3
The area of each side of a cube is s2. Since a cube has six square-shape sides,
its total surface area is 6 times s2.
Surface area of a cube = 6s2
Rectangular Prisms or Cuboids
A rectangular prism is also called a rectangular solid or a cuboid. In a
rectangular prism, the length, width and height may be of different lengths.

The volume of the above rectangular prism would be the product of the length,
width and height that is
Volume of rectangular prism = lwh
Total area of top and bottom surfaces is lw + lw = 2lw
Total area of front and back surfaces is lh + lh = 2lh
Total area of the two side surfaces is wh + wh = 2wh
Surface area of rectangular prism = 2lw + 2lh + 2wh = 2(lw + lh + wh)
Prisms
A prism is a solid that has two congruent parallel bases that are polygons. The
polygons form the bases of the prism and the length of the edge joining the
two bases is called the height.

Triangle-shaped base Pentagon-shaped base

The above diagrams show two prisms: one with a triangle-shaped base called a
triangular prism and another with a pentagon-shaped base called a pentagonal
prism.
A rectangular solid is a prism with a rectangle-shaped base and can be called a
rectangular prism.
The volume of a prism is given by the product of the area of its base and its
height.
Volume of prism = area of base × height
The surface area of a prism is equal to 2 times area of base plus perimeter of
base times height.
Surface area of prism = 2 × area of base + perimeter of base × height
Cylinders
A cylinder is a solid with two congruent circles joined by a curved surface.

In the above figure, the radius of the circular base is r and the height is h. The
volume of the cylinder is the area of the base × height.
Volume of cylinder = πr2h
The net of a solid cylinder consists of 2 circles and one rectangle. The curved
surface opens up to form a rectangle.

Surface area = 2 × area of circle + area of rectangle

Surface area of cylinder = 2πr2 + 2πrh = 2πr (r + h)
Spheres
A sphere is a solid with all its points the same distance from the center.

Cones
A circular cone has a circular base, which is connected by a curved surface to its
vertex. A cone is called a right circular cone, if the line from the vertex of the
cone to the center of its base is perpendicular to the base.

The net of a solid cone consists of a small circle and a sector of a larger circle.
The arc of the sector has the same length as the circumference of the smaller
circle.

Surface area of cone = Area of sector + area of circle

= πrs + πr2 = πr(r + s)
Pyramids
A pyramid is a solid with a polygon base and connected by triangular faces to
its vertex. A pyramid is a regular pyramid if its base is a regular polygon and
the triangular faces are all congruent isosceles triangles.

Nets Of A Solid
An area of study closely related to solid geometry is nets of a solid. Imagine
making cuts along some edges of a solid and opening it up to form a plane
figure. The plane figure is called the net of the solid.
The following figures show the two possible nets for the cube.