Welcome to the geometry section of our comprehensive ParaPro study guide. Geometry is a foundational branch of mathematics that you must be familiar with to succeed on the official ParaPro test. Below, our study guide covers the most common geometrical shapes as well as the terms and equations that are used to evaluate those shapes.
Geometrical Shapes
We will first go over some basic geometrical definitons and shapes.
- Perimeter: The sum of a shape’s side lengths.
- Area: The amount of space within a shape. The area will be represented in units\(^2\) because it is communicating two dimensions of space.
- 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: The perimeter of a circle.
Square:
Perimeter = \(4x\)
Area \(= x^2\)
Rectangle:
Perimeter = $2x + 2y$
Area = $xy$
Example 1
Find the area and perimeter of the rectangle.
Area \( = \) length \( \times \) width \( = 25 * 10 = 250\) cm\(^2\)
Perimeter \(= 25 + 10 + 25 + 10 = 70\) cm
Circle:
Circumference: \(πd \; (\text{or } 2πr)\)
Area \(= πr^2\)
Example 2
Find the area and circumference of the circle.
Area \( = π(4)^2 = 16 \ π\)
Diameter \(= 2 ∗ 4 = 8 \) cm
⇒ Circumference \(= 8 \ π\) cm
Example 3
Find the diameter of a basketball hoop that has a circumference of 56.54 inches.
Circumference \(= π \ \times \) diameter
\(⇒ 56.54 = π \ \times \) diameter
\(⇒ \dfrac{56.54}{π} = 18\) inches
Example 4
The circumference of a circle is 16π inches. What is the area of the circle?
Step 1: Use the formula for the circumference of the circle to find the radius of the circle.
\(C = 2πr\)
\(⇒ 16π = 2πr\)
\(⇒ \dfrac{16π}{2π} = r\)
\(⇒ 8 = r\)
Step 2: Find the area of the circle.
Area \(= 2πr^2\)
\(⇒\) Area \( = 2π(8)^2\)
\(⇒\) Area \(= 2π64\)
\(⇒\) Area \(= 128π\) inches
Parallelogram:
A parallelogram has two parallel sets of sides, but its corners do not have to be at right angles. Thus, squares and rectangles are technically parallelograms, but not all parallelograms are either squares or rectangles, such as the example below.
Area \(= base × height\)
Example 5
Find the height of a parallelogram that has an area of 52 units and base of 13 units.
Area of a parallelogram = base × height
\(⇒ 52 = 13 × \) height
\(⇒ \dfrac{52}{13} = 4\)The height of the parallelogram is \(4\) units.
Triangles:
Area \(= \dfrac{1}{2}\) base × height
Triangles, of course, are shapes with three sides. The same area formula is used for all triangles, but there are a few specific kinds of triangles that we’ve highlighted below.
Isosceles Triangles
This is a special triangle that has two congruent sides, meaning they are the same length. They are often shown marked with a straight line on the congruent sides.
Right Triangles
Right triangles have one right angle, which is usually shown with a square to indicate 90°.
You should now be able to solve problems based on different geometric shapes. To test your knowledge and ability to apply it on the official ParaPro test, use our quick, 5-question geometry review test below.
