Part of step 20 of 20·Geometry and Trigonometry·Area and Volume Prev

Math

Area and Perimeter

4 min readEasy10-question checkpoint

Area and perimeter questions are formula plug-ins — but the SAT loves to combine shapes (a rectangle with a semicircle on top, a triangle inside a rectangle) so you have to break compound figures into parts.

Where this lesson fits

Before

Right Triangles→

Area and Perimeter

You are here

→Volume →Circles

Perimeter is the distance around the outside of a 2D shape. Area is the space inside it.

The basic formulas to memorize:

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Common 2D shapes — area and perimeter

Shape	Area	Perimeter / Circumference
Rectangle (l × w)	$lw$	$2l + 2w$
Square (side $s$)	$s^2$	$4s$
Triangle (base $b$, height $h$)	$\frac{1}{2}bh$	Sum of three sides
Parallelogram	$bh$	Sum of four sides
Trapezoid (bases $b_1$, $b_2$)	$\frac{1}{2}(b_1 + b_2)h$	Sum of four sides
Circle (radius $r$)	$\pi r^2$	$2\pi r$

Rectangle: $A = lw$ (length × width). $P = 2l + 2w$ .

Square (special rectangle): $A = s^2$ . $P = 4s$ .

Triangle: $A = \frac{1}{2}bh$ (one-half base times height). The height must be PERPENDICULAR to the base, not necessarily one of the sides.

Parallelogram: $A = bh$ . The height is the perpendicular distance between the parallel base sides.

Trapezoid: $A = \frac{1}{2}(b_1 + b_2)h$ . Average of the two parallel bases × the height between them.

Circle: $A = \pi r^2$ . Circumference (perimeter): $C = 2\pi r$ .

Compound shapes. When the SAT shows a weird shape (e.g., a rectangle with a semicircular cutout):

Decompose into known shapes.
Compute area / perimeter of each part.
Add or subtract as the figure dictates.

Example: A rectangle 10 × 6 with a semicircle of radius 3 cut from one short end. Area = $10 \cdot 6 - \frac{1}{2}\pi(3)^2 = 60 - 4.5\pi$ .

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How to handle a compound shape

Is the shape a single recognized figure (rectangle / circle / triangle)?

Yes ↓

→ Use the standard formula directly

No ↓

Can you DECOMPOSE the shape into pieces (e.g., rectangle + semicircle)?

Yes ↓

Are pieces ADDED to make the shape, or SUBTRACTED (cut out)?

Yes ↓

→ ADD if the pieces combine to make the shape; SUBTRACT if part is cut away

No ↓

→ Compute each piece, then choose the right operation based on the figure

No ↓

→ Look for the OUTER region minus an INNER region — most compound figures are one shape with another removed

Common SAT trick: the area asked for might be a shaded region — the difference between two shapes.

Example: a circle inscribed in a square. Shaded = square area − circle area.

Perimeter pitfalls:

The perimeter of a compound shape is the OUTER boundary only. Internal lines that are shared between sub-shapes don't count.
Don't include curved parts where they've been cut off — only the part that's actually on the boundary.

Heron's formula (rare but useful): area of a triangle with sides $a, b, c$ when you don't know the height:

A = \sqrt{s(s-a)(s-b)(s-c)} \text{ where } s = \frac{a+b+c}{2}

$s$ is the semi-perimeter. Most SAT problems give you a base and height, but Heron's helps when only sides are given.

SAT-typical setups:

"What is the area of triangle ABC?" — find a base and the perpendicular height.
"What is the area of the shaded region?" — compute the larger area, subtract the unshaded piece.
"What is the perimeter?" — add only the outer-boundary segments.

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Quick check

Decompose compound shapes into known pieces. Compute each piece. Add or subtract per the figure. Watch radius/diameter and remember the $\frac{1}{2}$ for triangles and trapezoids.

A rectangular garden has a length of 12 meters and a width of 5 meters. What is the area, in square meters, of the garden?

Your answer:

Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).

Worked examples

Example 1

A rectangle has length 12 and width 8. A semicircle of diameter 8 is cut from one of the shorter sides. What is the area of the remaining shape?

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Example 2

A trapezoid has parallel bases of length 8 and 14, and a height of 5. What is its area?

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Common pitfalls

Using the wrong height for a triangle

Triangle area uses the PERPENDICULAR height to the chosen base. If the triangle is acute and the height isn't drawn, you may need to compute it (often via Pythagoras or special-triangle ratios). Don't use a side that isn't perpendicular to your base.

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Including internal segments in perimeter

If a compound shape has an internal line where two sub-shapes meet, that line is NOT part of the perimeter. Only the outer boundary counts.

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Confusing diameter with radius in circle areas

$A = \pi r^2$ uses radius. If you're given the diameter, divide by 2 first. Squaring the diameter gives an answer 4 times too big.

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Forgetting the half on triangle / trapezoid formulas

Triangle: $\frac{1}{2}bh$ . Trapezoid: $\frac{1}{2}(b_1 + b_2)h$ . Without the half, you double the answer. Same for the cone / pyramid volume formulas (which use $\frac{1}{3}$ ).

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Key takeaways

Rectangle: $A = lw$ . Square: $A = s^2$ . Triangle: $A = \frac{1}{2}bh$ .
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Parallelogram: $A = bh$ (perpendicular height). Trapezoid: $A = \frac{1}{2}(b_1 + b_2)h$ .
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Circle area: $\pi r^2$ . Circumference: $2\pi r$ .
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Compound shapes: decompose into parts, add or subtract.
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Perimeter = outer boundary only; internal shared lines don't count.
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Where this lesson fits

Before

Right Triangles→

Area and Perimeter

You are here

→Volume →Circles