ParaPro Math Study Guide: Geometry

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:

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:

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.

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 16π inches.  What is the area of the circle?

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 \(= 256π\)


Geometry Review Test


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