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Perimeter and Area

20 min Geometry

Perimeter and Area of Common Shapes

Perimeter measures the total distance around a shape, while area measures the space enclosed inside it. These two concepts are among the most practical in all of mathematics — you use them every time you buy fencing for a yard, paint a wall, or lay carpet in a room.

Each common shape — rectangles, triangles, parallelograms, trapezoids, and circles — has its own area formula, but they are all related. The area of a triangle is half the area of a rectangle, and the area of a trapezoid is the average of the two bases times the height.

This lesson covers perimeter and area formulas for all standard two-dimensional shapes, along with composite-figure problems that combine multiple shapes.

Formulas

Shape	Perimeter	Area
Rectangle	$P = 2l + 2w$	$A = lw$
Square	$P = 4s$	$A = s^2$
Triangle	$P = a + b + c$	$A = \frac{1}{2}bh$
Parallelogram	$P = 2a + 2b$	$A = bh$
Trapezoid	Sum of sides	$A = \frac{1}{2}(b_1 + b_2)h$

Example 1

Find the area and perimeter of a rectangle with $l = 15$ cm, $w = 8$ cm.

$A = 15 \times 8 = 120$ cm²

$P = 2(15) + 2(8) = 46$ cm

Example 2

Find the area of a triangle with base 14 and height 9.

$$A = \frac{1}{2}(14)(9) = 63$$

Example 3

A trapezoid has parallel sides 10 and 16, with height 7. Find the area.

$$A = \frac{1}{2}(10 + 16)(7) = \frac{1}{2}(26)(7) = 91$$

Example 4

A room is 12 ft × 15 ft. Carpet costs $\$4.50$ per square foot. Find the cost.

$A = 12 \times 15 = 180$ ft². Cost $= 180 \times \$4.50 = \$810$.

Circles

Circle Formulas

Circumference: $C = 2\pi r = \pi d$
Area: $A = \pi r^2$

Example 5

Find the circumference and area of a circle with radius 7.

$C = 2\pi(7) = 14\pi \approx 43.98$

$A = \pi(7)^2 = 49\pi \approx 153.94$

Example 6

A circular garden has diameter 20 m. Find the area.

$r = 10$. $A = \pi(10)^2 = 100\pi \approx 314.16$ m².

Practice Problems

1. Perimeter and area of a rectangle: $l=20$, $w=12$.

2. Area of a triangle: base $= 18$, height $= 11$.

3. Area of a trapezoid: bases $8$ and $14$, height $6$.

4. A square has perimeter 52. Find its area.

5. Circumference of a circle with $r = 5$.

6. Area of a circle with diameter 12.

7. A parallelogram has base 16 and height 9. Find the area.

8. A rectangular yard is 50 ft × 30 ft. Fencing costs $\$8.25$/ft. Find the fencing cost.

9. Find the area of a semicircle with radius 6.

10. A triangle has sides 5, 12, 13. Is it a right triangle? Find its area.

11. A running track is shaped like a rectangle (100 m × 50 m) with semicircles on each short end. Find the total perimeter.

12. Find the area of a square with diagonal $10\sqrt{2}$.

13. A pizza has diameter 16 inches. Find the area of one slice (8 equal slices).

14. Area of an equilateral triangle with side 10. (Use $A = \frac{\sqrt{3}}{4}s^2$.)

15. A rectangle has area 120 and width 8. Find the perimeter.

Show Answer Key

1. $P = 64$; $A = 240$

2. $99$

3. $66$

4. Side $= 13$; $A = 169$

5. $10\pi \approx 31.42$

6. $r = 6$; $A = 36\pi \approx 113.10$

7. $144$

8. $P = 160$ ft; Cost $= \$1{,}320$

9. $\frac{1}{2}\pi(36) = 18\pi \approx 56.55$

10. Yes ($25 + 144 = 169$). $A = \frac{1}{2}(5)(12) = 30$

11. $2(100) + \pi(50) \approx 357.08$ m

12. Side $= 10$; $A = 100$

13. $\frac{\pi(8)^2}{8} = 8\pi \approx 25.13$ sq. in.

14. $\frac{\sqrt{3}}{4}(100) = 25\sqrt{3} \approx 43.30$

15. $l = 15$; $P = 2(15)+2(8) = 46$

Angles, Lines, and Triangles Surface Area and Volume

Perimeter and Area