## Geometrical Shapes

We will first go over some basic geometrical definitons and shapes.

- Perimeter: the sum of a shape’s side lengths.
- Diameter (\(\,d \,\)): the distance of a line segment that passes through the center and has 2 endpoints on the edge of the circle; twice the radius.
- Radius (\(\,r \,\)): any segment from the center of a circle to its perimeter; half of the diameter.
- Circumference: perimeter of a circle.

## Circle:

Circumference: \(πd \; (\text{or } 2πr)\)

Area \(= πr^2\)

## Square:

Perimeter: \(4x\)

Area \(= x^2\)

## Rectangle:

Perimeter = $2x + 2y$

Area = $xy$

## Parallelogram: two sets of parallel sides

Area \(= base × height\)

## Triangles:

Area \(= \dfrac{1}{2} base × height\)

## Isosceles Triangles: two congruent sides

## Right Triangles: One Right Angle

Example 1

### Find the area and circumference of the circle.

*Solution:*

Using the formula for area of a circle, we compute

Area \( = π(4)^2 = 16 \π\).

To find circumference, we computer \( π x \) the diameter of the circle:

Diameter \(= 2 ∗ 4 = 8 \ cm\)

⇒ Circumference \(= 8 \ π\)

To calculate the area of a rectangle, you multiply the length \( x \) width.

Example 2

**Find the area and perimeter of the rectangle.**

*Solution:*

*Solution:*

Area \( = \) length \( \times \) width \( = 25 * 10 = 250 \ cm^2\)

Perimeter \(= 25 + 10 + 25 + 10 = 70 \ cm\)

If you are given the circumference of a circle, you can use the formula for circumference, \(C = π d\), to find the diameter or radius.

Example 3

**Find the diameter of a basketball hoop that has a circum- ference of 56.54 inches.**

*Solution:*

*Solution:*

Circumference \(= π \ \times \) diameter

\(⇒ 56.54 = π \ \times \) diameter

\(⇒ \dfrac{56.54}{π} = 18\) inches.

Example 4

**Find the height of a parallelogram that has an area of 52 and base of 13.**

**Find the height of a parallelogram that has an area of 52 and base of 13.**

*Solution:*

*Solution:*

Area of a parallelogram = base × height

\(⇒ 52 = 13 × \) height

\(⇒ \dfrac{52}{13} = 4.\)So, the height of the parallelogram is \(4\).

Example 5

**The circumference of a circle is 16***π ***inches. What is the area of the circle?**

**The circumference of a circle is 16***π***inches. What is the area of the circle?***Solution:*

*Solution:*

Step 1: Use the a for circumference of the circle to find the radius of the circle.

\(C = π d = 2πr\)

\(⇒ 32π = 2πr\)

\(⇒ \dfrac{32π}{2π} = r.\)

\(⇒ 16 = r.\)

Step 2: Find the area of the circle.

Area \(= 2πr^2\)

\(⇒\) Area \( = 2π(16)^2\)

\(⇒\) Area \(= 256π\)