Area and Perimeter of Basic Shapes
SAT area and perimeter questions like to combine shapes: a rectangle with a semicircle on one end, a ring between two circles, a square around a circle. Others ask what happens to the area when every side is scaled, or how to convert square feet to square yards.
Where I would start
Area is the space inside a shape, measured in square units. Perimeter is the distance around it, in plain units. For an unusual shape, split it into rectangles, triangles and pieces of circles, then add or subtract.
The reference sheet gives you the area of a circle, a rectangle and a triangle, and the circumference of a circle. It does not give the trapezoid, and it won't tell you which formula a question needs.
Formulas and question types
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | $A = \ell w$ | $P = 2\ell + 2w$ |
| Triangle | $A = \dfrac{1}{2}bh$ ($h$ is perpendicular to $b$) | Add the three sides |
| Circle | $A = \pi r^2$ | $C = 2\pi r = \pi d$ |
| Parallelogram | $A = bh$ | Add the four sides |
| Trapezoid | $A = \dfrac{1}{2}(b_1 + b_2)h$ | Add the four sides |
- Scaling: if every length is multiplied by $k$, the perimeter is multiplied by $k$ and the area by $k^2$.
- Square units: $1$ foot $= 12$ inches, so $1$ square foot $= 12^2 = 144$ square inches. $1$ yard $= 3$ feet, so $1$ square yard $= 9$ square feet.
Examples
The dashed line is inside the figure, so it is not part of the perimeter.
A rectangle with a semicircle
The figure shows a rectangle $20$ units long and $8$ units wide, with a semicircle on one short side. Find the area and the perimeter.
- The semicircle's diameter is the short side, $8$, so its radius is $4$.
- Area: rectangle $20 \cdot 8 = 160$, plus half a circle $\dfrac{1}{2}\pi(4)^2 = 8\pi$. Total $160 + 8\pi$.
- Perimeter: three sides of the rectangle, $20 + 8 + 20 = 48$, plus half the circumference, $\dfrac{1}{2} \cdot 2\pi(4) = 4\pi$. Total $48 + 4\pi$. The dashed side is not on the outside, so it is not counted.
The ring between two circles
Two circles share a center. The smaller has radius $7$ and the larger has radius $10$. What is the area of the region between them?
- Larger circle: $\pi(10)^2 = 100\pi$. Smaller circle: $\pi(7)^2 = 49\pi$.
- Subtract: $100\pi - 49\pi = 51\pi$.
- The answer is not $\pi(10 - 7)^2 = 9\pi$. You subtract the two areas, not the two radii.
Scaling a shape
A rectangle has an area of $30$ square units. Each side is tripled. What is the new area?
- Every length is multiplied by $3$, so the area is multiplied by $3^2 = 9$.
- $30 \cdot 9 = 270$ square units.
- Check with a $5$ by $6$ rectangle: the new one is $15$ by $18$, and $15 \cdot 18 = 270$.
Converting square units
A room is $12$ feet by $15$ feet. Carpet is sold by the square yard. How many square yards cover the floor?
- Area in square feet: $12 \cdot 15 = 180$.
- One square yard is $9$ square feet, so divide by $9$: $180 \div 9 = 20$ square yards.
- Or convert first: the room is $4$ yards by $5$ yards, which is $20$ square yards.
Where students lose points
- Using a slanted side as the height. The height of a triangle or parallelogram must be perpendicular to the base.
- Counting an inside edge in the perimeter. When shapes are joined, the shared side is no longer on the outside.
- Mixing up radius and diameter. If a semicircle sits on a side of length $8$, its radius is $4$.
- Scaling area by $k$ instead of $k^2$. Doubling every side makes the area four times as large.
- Converting square units with the linear factor. Square feet to square inches uses $144$, not $12$.
Practice
Draw each figure and label it before you start. Then open the solution to compare.
Easy
A rectangle has an area of $48$ square units and a length of $8$ units. What is its perimeter, in units?
- A$14$
- B$28$
- C$48$
- D$56$
Show solution
Answer: B. The width is $48 \div 8 = 6$. The perimeter is $2(8) + 2(6) = 28$. Choice A adds the length and width only once.
Easy
A circle has a diameter of $10$ units. What is its area, in square units?
- A$10\pi$
- B$20\pi$
- C$100\pi$
- D$25\pi$
Show solution
Answer: D. The radius is $5$, so the area is $\pi(5)^2 = 25\pi$. Choice C uses the diameter in place of the radius.
MediumStudent-produced response
A right triangle has legs of length $9$ and $12$. What is the perimeter of the triangle?
Enter your answer.
Show solution
Answer: $36$. The hypotenuse is $\sqrt{9^2 + 12^2} = \sqrt{225} = 15$. The perimeter is $9 + 12 + 15 = 36$.
Medium
Two circles have the same center $O$. Point $B$ is on the smaller circle and point $C$ is on the larger circle, with $OB = 7$ and $OC = 10$. What is the area of the region inside the larger circle and outside the smaller circle?
- A$3\pi$
- B$9\pi$
- C$51\pi$
- D$149\pi$
Show solution
Answer: C. $\pi(10)^2 - \pi(7)^2 = 100\pi - 49\pi = 51\pi$. Choice B squares the difference of the radii, and choice D adds the two areas.
Medium
A figure is made of a rectangle with length $20$ and width $8$ and a semicircle whose diameter is one of the rectangle's shorter sides. What is the area of the figure?
- A$160 + 4\pi$
- B$160 + 16\pi$
- C$160 + 64\pi$
- D$160 + 8\pi$
Show solution
Answer: D. The semicircle has radius $4$, so its area is $\dfrac{1}{2}\pi(4)^2 = 8\pi$. The rectangle is $160$. Total: $160 + 8\pi$. Choice B uses a full circle.
MediumStudent-produced response
A square has an area of $25$ square units. Each side of the square is multiplied by $3$. What is the area of the new square, in square units?
Enter your answer.
Show solution
Answer: $225$. The side goes from $5$ to $15$, so the new area is $15^2 = 225$. In general, multiplying lengths by $3$ multiplies the area by $3^2 = 9$, and $25 \cdot 9 = 225$.
Hard
An isosceles right triangle has a hypotenuse of $v$ feet. Which expression represents the perimeter of the triangle, in feet?
- A$v + v\sqrt{2}$
- B$2v + v\sqrt{2}$
- C$3v$
- D$\dfrac{v + v\sqrt{2}}{2}$
Show solution
Answer: A. Each leg is $\dfrac{v}{\sqrt{2}} = \dfrac{v\sqrt{2}}{2}$. Two legs add to $v\sqrt{2}$, and the hypotenuse adds $v$. The perimeter is $v + v\sqrt{2}$. Choice B treats each leg as $v$.
Hard
A room is $12$ feet by $15$ feet. Carpet costs \$25 per square yard. What is the cost of carpet to cover the whole floor? ($1$ yard $= 3$ feet)
- A\$500
- B\$1,500
- C\$4,500
- D\$166.67
Show solution
Answer: A. The area is $12 \cdot 15 = 180$ square feet, and $180 \div 9 = 20$ square yards. The cost is $20 \cdot 25 = 500$ dollars. Choice B divides by $3$ instead of $9$, and choice C forgets to convert.
HardStudent-produced response
A trapezoid has parallel sides of length $6$ and $14$ and a height of $5$. What is the area of the trapezoid?
Enter your answer.
Show solution
Answer: $50$. $A = \dfrac{1}{2}(6 + 14)(5) = \dfrac{1}{2}(20)(5) = 50$.
Hard
A circle is inscribed in a square with sides of length $10$, so the circle touches all four sides. What is the area of the region inside the square but outside the circle?
- A$100 - 100\pi$
- B$100 - 25\pi$
- C$100 - 10\pi$
- D$75\pi$
Show solution
Answer: B. The circle's diameter equals the side of the square, $10$, so its radius is $5$ and its area is $25\pi$. The square's area is $100$. The region between them is $100 - 25\pi$, about $21.5$. Choice A uses $10$ as the radius.
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