Geometry Basics — Core Skills

Geometry Basics: Area and Perimeter

Part 1 of 2 — One Skill, One Idea

Two words come up over and over, and they mean different things.

• Perimeter is the distance all the way around the outside. Think of walking around the edge of a yard. It is measured in plain units, like feet.
• Area is how much flat space is inside. Think of the carpet that covers the floor. It is measured in square units, like square feet.

If a question asks how much fence you need, that is perimeter. If it asks how much paint or carpet, that is area.

The formulas you need

• Rectangle area $= \text{length} \times \text{width}$
• Rectangle perimeter $= 2 \times \text{length} + 2 \times \text{width}$
• Square area $= \text{side} \times \text{side}$
• Square perimeter $= 4 \times \text{side}$
• Triangle area $= \frac{1}{2} \times \text{base} \times \text{height}$

A square is a rectangle whose four sides are all the same length, so the rectangle formulas still work on it.

Worked example

A rectangle has length $8$ and width $5$. Find the area and the perimeter.

Area — multiply the two side lengths:

$8 \times 5 = 40$

The area is $40$ square units.

Perimeter — add up all four sides. The rectangle has two sides of length $8$ and two sides of length $5$:

$8 + 8 + 5 + 5 = 26$

The perimeter is $26$ units.

Notice that the two answers are different numbers from the same rectangle. Read the question and decide which one it wants before you start.

Angles and Right Triangles

Part 2 of 2 — Practice

The formulas from Part 1

• Rectangle area $= \text{length} \times \text{width}$
• Square area $= \text{side} \times \text{side}$
• Triangle area $= \frac{1}{2} \times \text{base} \times \text{height}$
• Perimeter $=$ add up every side

Angle facts

These four cover most SAT angle questions.

• Straight line: $180^\circ$. Two angles that together form a straight line are called supplementary. They add to $180^\circ$.
• Right angle: $90^\circ$. Two angles that together form a right angle are called complementary. They add to $90^\circ$.
• Triangle: $180^\circ$. The three angles inside any triangle add to $180^\circ$.
• Vertical angles are equal. When two straight lines cross, they make an X. The two angles directly across from each other are equal.

To find a missing angle, write the total and subtract what you know. If two angles are supplementary and one is $130^\circ$, the other is $180 - 130 = 50^\circ$.

The Pythagorean theorem

A right triangle has one $90^\circ$ angle. The two short sides that form the right angle are the legs. The long side across from the right angle is the hypotenuse.

$a^{2} + b^{2} = c^{2}$

Here $a$ and $b$ are the legs and $c$ is the hypotenuse. The hypotenuse is always alone on its own side of the equal sign.

Worked example. Legs $3$ and $4$:

$3^{2} + 4^{2} = 9 + 16 = 25$

$c = \sqrt{25} = 5$

Three sets of numbers show up constantly: $3$-$4$-$5$, $6$-$8$-$10$, and $5$-$12$-$13$. Knowing them saves time.