Area and Perimeter of Basic Shapes
Area measures the space inside; perimeter measures the distance around. Composite figures just break into shapes you already know.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
Use the right formula for each shape, and read carefully whether the question wants area or perimeter. For a composite figure, split it into rectangles, triangles, and circles, then combine.
How to solve one, step by step
Example: a rectangle $5$ by $3$.
- Area: $5 \times 3 = 15$ square units.
- Perimeter: $2(5) + 2(3) = 16$ units.
The mistakes that cost points
- Computing area when perimeter is asked. Check which the question wants — they're easy to swap.
- Not breaking up a composite figure. Split it into familiar shapes and add (or subtract) the pieces.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
The figure shows two concentric circles with center $O$. The length of $OB$ is $7$ units and the length of $OC$ is $10$ units. What is the area of the shaded region, in square units?
- A$3\pi$
- B$149\pi$
- C$51\pi$
- D$9\pi$
Show solution
Answer: C, $51\pi$. $OC$ is the radius of the outer circle, so the outer circle has area $\pi (OC)^2 = \pi (10)^2 = 100\pi$ square units. $OB$ is the radius of the inner circle, so the inner circle has area $\pi (OB)^2 = \pi (7)^2 = 49\pi$ square units. The shaded region is the area between the two circles, so we subtract: $100\pi - 49\pi = 51\pi$ square units.
Medium
The figure shows a rectangle with a semicircle attached to one of its shorter sides. The rectangle has a length of $20$ units and a width of $6$ units. What is the total area of the figure, in square units?
- A$120$
- B$\frac{240 + 9\pi}{2}$
- C$120 + 18\pi$
- D$120 + 9\pi$
Show solution
Answer: B, $\frac{240 + 9\pi}{2}$. The area of the rectangle is $20 \times 6 = 120$ square units. The semicircle is attached to the shorter side of length $6$, so its diameter is $6$ and its radius is $r = \frac{6}{2} = 3$. The area of a semicircle is $\frac{1}{2}\pi r^2 = \frac{1}{2}\pi (3)^2 = \frac{9\pi}{2}$ square units. Combining with a common denominator: $120 + \frac{9\pi}{2} = \frac{240}{2} + \frac{9\pi}{2} = \frac{240 + 9\pi}{2}$ square units.
Hard
An isosceles right triangle has a hypotenuse of $v$ feet. Which expression represents the perimeter, in feet, of this triangle?
- A$v + v\sqrt{2}$
- B$v\sqrt{2}$
- C$v + 2v\sqrt{2}$
- D$3v$
Show solution
Answer: A, $v + v\sqrt{2}$. STEP 1: Identify the triangle: isosceles right means two equal legs, with a $90°$ angle and a hypotenuse opposite it.
STEP 2: The hypotenuse is $v$.
STEP 3: Each leg has length $\frac{v\sqrt{2}}{2}$.
STEP 4: Perimeter = leg + leg + hypotenuse = $\frac{v\sqrt{2}}{2} + \frac{v\sqrt{2}}{2} + v = v + v\sqrt{2}$ feet.
Find your exact gaps
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