SHSAT: Area of composite shapes
Find areas of L-shapes, borders and cut-outs by splitting them into rectangles, triangles, trapezoids and circles.
How do you find the area of a shape that isn't a simple rectangle?
Short answer
Split the shape into pieces whose formulas you know. Add the pieces, or find the whole and subtract what's cut away.
Know first: Multiplying whole numbers and decimals, Rectangle and triangle area
Step 1 of 5
Split a strange shape into shapes you know
Most area questions use a shape you can't find with one formula: an L-shape, a frame, a square with a circle cut out. Break it into rectangles, triangles, trapezoids and circles. Then add the pieces or subtract the hole. If two pieces overlap, add their areas and subtract the overlap once: two 30-square-foot mats that overlap by 8 square feet cover 52 square feet.
Area is measured in square units and perimeter in plain units. A perimeter can't be turned straight into an area, and a slanted side is not a height.
| Shape | Area | Watch out |
|---|---|---|
| Rectangle | length × width | perimeter is 2(length + width), a different measure |
| Triangle | ½ × base × height | height is perpendicular to the base |
| Trapezoid | ½ × (base 1 + base 2) × height | the bases are the parallel sides |
| Circle | π × radius² | halve the diameter first |
The SHSAT guide says formulas won't be supplied, so learn these with their conditions.
Step 2 of 5
Add pieces, subtract holes
Worked example: A rectangle with a corner cut out
A 15-by-8 rectangle has a 6-by-3 rectangle cut from one corner. What is the area that remains, in square units?
- A18
- B46
- C102
- D120
- 1
Area of the full rectangle
15 × 8 = 120.
- 2
Area of the cut-out
6 × 3 = 18.
- 3
Subtract
120 − 18 = 102 square units.
- 4
See why the other choices are there
18 is the piece removed. 46 is the perimeter of the full rectangle, a length. 120 forgets to remove the corner.
Answer
102 square units
Wrong: A garden is 22 by 16 meters, with a path 3 meters wide inside all four edges. The inner rectangle is 19 by 13.
Right: The path takes 3 meters from both ends of each side, so the inner rectangle is 16 by 10. Path area = 352 − 160 = 192 square meters.
A border along all four edges shortens each dimension twice, once at each end.
A circle cut from a square
A circle of radius 4 is cut from a square of side 10. Use π = 3.14.
Square: 10 × 10 = 100
Circle: 3.14 × 4² = 3.14 × 16 = 50.24
Remaining: 100 − 50.24 = 49.76 square units
Step 3 of 5
Find the right measurements first
Check yourself · Question 1
A triangle has a base of 18 centimeters, a perpendicular height of 8 centimeters, and a slanted side of 11 centimeters. What is its area in square centimeters?
Answer: A
Use the height that meets the base at a right angle: ½ × 18 × 8 = 72 square centimeters. The slanted side is not used.
Trap answer: 99
Using the slanted side
- Why it looks right
- 11 is labeled on the triangle and looks like a height going up from the base.
- Why it's wrong
- Height must be perpendicular to the base. A slanted side is longer than the height, so the area comes out too big.
- The right answer
- 72, from ½ × 18 × 8.
Check yourself · Question 2
A trapezoid has parallel sides of 9 meters and 17 meters, and a height of 6 meters. What is its area in square meters?
Answer: B
Average the parallel sides: (9 + 17) ÷ 2 = 13. Multiply by the height: 13 × 6 = 78 square meters.
Check yourself · Question 3
A rectangle's length increases by 25% and its width decreases by 20%. The new area is what percent of the old area?
Answer: B
Multiply the factors: 1.25 × 0.8 = 1. The new area is 100% of the old area, so the area doesn't change.
Common mistake
I add the percent changes in length and width to get the change in area.
- Why it's tempting
- Adding is what you'd do if the two changes applied to the same number.
- Do this instead
- Area is length times width, so the changes multiply. Turn each into a factor, such as 1.25 and 0.8, and multiply.
Step 4 of 5
What to remember
Remember
- Break a composite shape into known shapes, then add the pieces or subtract the holes.
- Use the perpendicular height for triangles and trapezoids, and halve a diameter before using πr².
- A border inside all edges shortens each side twice, and percent changes in dimensions multiply.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, what were the two areas you found, and why did you subtract rather than add?