**Circumference and Area of Circles, Pi, Perimeter and Area of Triangles/Rectangles**

**What is a Circle?**

**What is a Circle?**

A circle is a shape where the *circumference *(or the perimeter of the circle) consists of points that are equally far away from a fixed point (the center).

The **radius** is the distance from the center of the circle to any point on the circle itself (must be from the center)

The **diameter** is the distance from one point on the circle to the other side (must go through the center)

The diameter is always twice the length of the radius.

**What is the Circumference a Circle?**

**What is the Circumference a Circle?**

The** circumference **is the distance around a circle (this is a special term for the perimeter of circles).

To find the circumference of a circle, you use the equation:

*C *= d*π**,* which is more commonly written as *C *= 2*πr*

**C = d****π ****= 2****πr**

**C = d**

**π**

**= 2**

**πr**

*Ex.1: What is the circumference of a circle in terms of pi with a radius of 12?*

*C *= 2*πr = *2*π(12) = 24π*

*Answer: 2**4π*

**What is the Area a Circle?**

**What is the Area a Circle?**

The **area** of a circle is the amount of space that the circle covers (total amount of space inside the circle)

To find the area of a circle, you use the equation:

Area of circle=*π *× radius^{2} , or

**A**** = ****πr**^{2}

**A**

**=**

**πr**

^{2}

*Ex**.** 1:* In terms of pi, what is the area of a circle with a radius of 5?

A = πr^{2} = π × (5)^{2} = 25π

*Answer: 25π = ~78.54*

*Ex**. **2: *What is the area of a circle in terms of pi with a diameter of 16?

Diameter = 2 × (Radius)

16 = 2 (Radius)

Radius = 8

Area = *π *× radius^{2} = *π *× (8)^{2} = 64*π*

*Answer: *64*π = ~201.16*

**What is Pi?**

**What is Pi?**

We have been using it a lot in equations involving circles lately, so what exactly is it? **Pi **is the ratio of any circle's circumference to its diameter (this is true for all circles!). If you have a circle with a diameter of 1, for example, then the circumference of the circle is 3.141592653…or pi.

Pi is written as the Greek symbol *π.*

From the drawing, the circumference divided by the diameter of the circle is π (this is where our equation for circumference comes from).

Finding the Circumference using the Diameter

*Ex**.** 1:* You walk around a circle which has a diameter of 100 meters, how far have you walked?

Distance walked = circumference

Circumference = πd

= π(100) = 314 m (don't forget units!)

Finding the Diameter using the Circumference

*Ex**.** 1:* Sam measured 80 mm around the outside of a pipe, what is its diameter?

Diameter = Circumference / π

= 80 mm / π

= 25.464

Answer: 25 mm (closest mm

**Pi is approximately equal to:**

**Pi is approximately equal to:**

**3.14159265358979323846264338327950…**

**3.14159265358979323846264338327950…**

A quick and easy approximation for π using fractions is 22/7

● 22/7 = 3.1428571…

○ Keep in mind that 22/7 is not exactly equal to π

○ π is not exactly equal to the ratio of any two whole numbers because π is an **irrational number**

● Better approximation: 355/113

○ 335/113 = 3.1415929…

When using π in calculations, usually 3.14 or 3.14159 is fine. Some calculators have a built in button for π that uses a very good approximation for the value (you can use this button instead if you want).

**What is a Triangle?**

**What is a Triangle?**

A **triangle** is a shape that has three sides and three angles.

1) The three angles always add up to 180 degrees.

2) There are 3 different types of triangles, with three different angle classifications:

You can also combine names together to create triangles. The triangle on the right is called an isosceles right triangle because it has a right angle and two equal angles.

What is the Perimeter of a Triangle?

The **perimeter** is the total distance around the edge of the triangle.To find the perimeter of a triangle, you simply add up all the side lengths (P = a + b + c).

*Ex**.** 1:* What is the perimeter of the triangle on the right?

Perimeter = 11cm + 13cm + 8cm = 32cm

What is the Area of a Triangle?

The area of a triangle is the total amount of space inside the triangle. Area = ½ x b x h , where b is the distance along the base and h is the height (measured at a right angle to the base).

## **A = ½ bh **

**A = ½ bh**

*Ex**.**1:* What is the area of the triangle to the right?

Area = ½ x b x h

= ½ x 30 x 21

= 315

**Answer: **Area = 315 units^{2}

**What is a Rectangle?**

**What is a Rectangle?**

A rectangle is a shape with four sides, where every angle is a right angle (90°). The opposite sides of a rectangle are equal to each other (the dashes indicate that they are equal)

What is the Perimeter of a Rectangle?

The perimeter of a rectangle is the distance around the edges. To find the perimeter, you can add up all the sides if they are given, or more commonly, use the equation: Perimeter = 2(w+h), where w is the width and h is the height of the rectangle. (you can also use length and width instead).

**P = 2(w + h)**

**P = 2(w + h)**

*E**x.** *1: A rectangle has a width of 10 cm, and a height of 6 cm, what is its perimeter?

Perimeter = 2 x (10cm + 6 cm)

= 2 x (16cm)

= 32 cm

**Answer: **Perimeter = 32 cm

What is the Area of a Rectangle?

The area of a rectangle is the amount of space it takes up, and is found using the equation: Area = w x h , where w is the width and h is the height.

**A = wh**

**A = wh**

*Ex**.** 1: A rectangle is 8m wide and 5m high, what is its area?*

Area = w x h

= 8m x 5m

= 40 m^{2}

**Answer:** Area = 40 m^{2}