Find The Perimeter And Area Of The Following Figures

Find the Perimeter and Area of the Following Figures

Ever stared at a geometry worksheet filled with shapes — rectangles, triangles, circles — and felt your brain go fuzzy when you hit the word problems? That said, you're not alone. Whether you're a student grinding through homework, a parent trying to help without waking up your old math textbooks, or someone just trying to figure out how much paint to buy for a room, calculating perimeter and area comes up way more often than you'd expect.

Here's the good news: once you understand the logic behind these formulas, they stick. And that's exactly what we're going to do right now.

What Are Perimeter and Area, Really?

Let's strip away the math-class language for a second.

Perimeter is the distance all the way around the outside of a shape. Imagine walking the boundary of a rectangular park — every step you take along the edge adds up. That total distance? That's your perimeter. You measure it in linear units like feet, meters, or centimeters.

Area is the amount of space inside a shape. If you wanted to cover that same rectangular park with sod, the area tells you how much sod you'd need. Area is measured in square units — square feet, square meters, and so on.

One reason people get mixed up? Perimeter and area are related, but they're not interchangeable. A shape can have a small area but a huge perimeter (think of a long, skinny rectangle). And vice versa — a compact shape can enclose a lot of space without much boundary. Keeping that distinction clear in your head makes everything else click.

How to Find Perimeter and Area of Common Figures

Here's where we get into the actual math. Let's walk through each figure, show you the formulas, and walk through an example so you can see how it all works in practice.

Rectangle

A rectangle is probably the most familiar shape you'll encounter. It has four right angles, with opposite sides that are equal in length.

Perimeter of a rectangle: Add up all four sides. Since opposite sides are equal, this simplifies to: P = 2(length + width)

Area of a rectangle: Multiply length by width: A = length × width

Example:* A rectangle has a length of 8 cm and a width of 5 cm.

• Perimeter = 2(8 + 5) = 2 × 13 = 26 cm
• Area = 8 × 5 = 40 cm²

Square

A square is just a rectangle where all four sides are the same length. That makes the formulas simpler.

Perimeter of a square: Since all sides are equal: P = 4 × side

Area of a square: A = side × side (or side²)

Example:* A square has a side length of 6 inches.

• Perimeter = 4 × 6 = 24 inches
• Area = 6 × 6 = 36 square inches

Triangle

Triangles come in different flavors — equilateral (all sides equal), isosceles (two sides equal), and scalene (no sides equal). But the area formula works for all of them. Perimeter is just adding the three side lengths.

Perimeter of a triangle: P = side₁ + side₂ + side₃

Area of a triangle: The classic formula: A = ½ × base × height

The base* is any one side you choose. The height* is the perpendicular distance from that base to the opposite vertex — it's not necessarily one of the triangle's sides unless it's a right triangle.

Example:* A triangle has a base of 10 cm and a height of 7 cm.

• Area = ½ × 10 × 7 = 35 cm²

If the sides measure 6 cm, 8 cm, and 10 cm, the perimeter is simply 6 + 8 + 10 = 24 cm.

Circle

Circles are a bit different because they don't have straight edges. So instead of perimeter, we call the distance around a circle the circumference. And calculating it involves a special number: pi (π), which is roughly 3.14159.

Circumference of a circle: C = 2πr where r is the radius (the distance from the center to any point on the edge)

You can also use the diameter: C = πd since the diameter is just 2r.

Area of a circle: A = πr²

Example:* A circle has a radius of 4 meters. Which means - Circumference = 2π(4) = 8π ≈ 25. 13 meters

• Area = π(4)² = 16π ≈ **50.

Most calculators have a π button. If you're working by hand and the problem doesn't specify, using 3.14 for π is usually fine.

Parallelogram

A parallelogram is a four-sided shape where opposite sides are parallel and equal — like a tilted rectangle. The base and height are perpendicular to each other, but the sides (the "slant") are different.

Continue exploring with our guides on the picture below shows the graph of which inequality -4 and 9x - 8y 12 - 8y.

Perimeter of a parallelogram: Add all four sides, or P = 2(base + slant height)

Area of a parallelogram: A = base × height

The height* is the perpendicular distance between the two parallel bases — not the slant side.

Example:* A parallelogram has a base of 9 cm and a height of 4 cm.

• Area = 9 × 4 = 36 cm²

Trapezoid

A trapezoid (called a trapezium in some countries) has one pair of parallel sides that are different lengths — these are the two bases. The non-parallel sides are the legs.

Perimeter of a trapezoid: Add all four sides: P = base₁ + base₂ + leg₁ + leg₂

Area of a trapezoid: A = ½ × (base₁ + base₂) × height

Example:* A trapezoid has bases of 6 cm and 10 cm, with a height of 4 cm.

• Area = ½ × (6 + 10) × 4 = ½ × 16 × 4 = 32 cm²

Common Mistakes to Watch Out For

Geometry problems have a way of tripping people up on a few predictable points. Here's where I see students (and plenty of adults) go wrong.

Mixing up perimeter and area formulas. This is the big one. People sometimes multiply when they should be adding, or vice versa. A quick mental check: perimeter is about distance (linear), area is about coverage (square). If your answer

If your answer should be in linear units (cm, m, in) and you're getting a squared value, something's off.

Using the slant side as the height. In triangles, parallelograms, and trapezoids, the height must be perpendicular to the base. That slanted side running along the edge of the shape is almost never the height. Read the problem carefully — if it gives you the slant but not the perpendicular height, you'll need extra information (like an angle) to find the true height.

Forgetting to halve the area in triangles and trapezoids. Students often compute base × height and forget the ½ in front. It's a small step but it doubles your answer, which is usually wrong.

Rounding π too early. If you're carrying a problem through multiple steps, keep π in your calculations as long as possible. Round only at the end, and only to the precision the problem asks for.

Units. Always include units in your final answer, and make sure they make sense. If the sides are in centimeters, the area is in square centimeters (cm²) and the perimeter is in centimeters (cm). Mixing units — like adding meters and centimeters without converting — is a common source of wrong answers.

Not drawing a picture. Even for simple problems, sketching the shape with the given measurements labeled can prevent silly mistakes. You might realize, for example, that the "base" given isn't the one you assumed, or that two sides are actually equal when you thought they weren't.

A Few Quick Practice Problems

Test yourself with these. The answers are below, but give them a real try first.

1. A rectangle is 12 cm long and 5 cm wide. Find its area and perimeter.
2. A triangle has a base of 14 cm and a height of 9 cm. Find its area.
3. A circle has a diameter of 10 inches. Find its circumference and area.
4. A parallelogram has a base of 11 m, a height of 6 m, and slanted sides of 7 m. Find its area and perimeter.
5. A trapezoid has parallel sides of 8 cm and 14 cm, a height of 5 cm, and legs of 6 cm each. Find its area and perimeter.

Answers:

1. Area = 60 cm²; Perimeter = 34 cm
2. Area = 63 cm²
3. Circumference = 10π ≈ 31.42 in; Area = 25π ≈ 78.54 in²
4. Area = 66 m²; Perimeter = 36 m
5. Area = 55 cm²; Perimeter = 34 cm

Final Thoughts

Perimeter and area are the foundation of nearly everything you'll do in geometry, from simple homework problems to real-world tasks like measuring a room for flooring or figuring out how much fencing a yard needs. The formulas themselves aren't difficult — the trick is keeping them straight and applying the right one at the right time.

A few habits that help:

• Write the formula first before plugging in numbers. This slows you down just enough to avoid the most common errors.
• Label your work with units at every step. It's a small thing, but it catches mistakes.
• Estimate the answer before you compute. If a rectangle is roughly 10 by 5, the area should be around 50, not 5 or 500. Sanity checks like this save you from bad answers that happen to look correct.

Once you're comfortable with these shapes, the same logic extends to more complex figures — and you'll find that the underlying ideas (perimeter is the boundary, area is the space inside) never really change. Master the basics, and the rest follows.