What is the difference between area and perimeter?

Perimeter is the distance around the outside — the fence. Area is the surface inside — the grass. Perimeter is measured in plain units, area in square units, and two shapes can share a perimeter while having very different areas.

How do I find the area of an irregular or composite shape?

Split it into standard shapes with a straight cut, find each area, then add them — or start from a bounding rectangle and subtract the missing piece. Both routes give the same answer; pick whichever needs fewer unknown lengths.

Do I use radius or diameter in the circle formula?

Radius, always. If the problem gives the diameter, halve it first. Using the diameter in \pi r^2 makes the area four times too large — it is the single most common circle mistake.

Should I leave \pi in the answer or use 3.14?

Leave it exact as 49\pi unless the question asks for a decimal or specifies a value of \pi. Exact answers avoid rounding errors, and most mark schemes accept both.

Can I find a missing side if I know the area?

Yes — substitute into the formula and solve the equation for the unknown. If the problem relates two sides in words, such as "the length is 3 more than the width", you will end up with a quadratic to factor.

Can Photo Math Solver read the shape from a diagram?

Yes — screenshot the figure with its labels, including the dashed perpendicular height if there is one, and the solver identifies the shape and picks the matching formula before substituting.

You can start solving area and perimeter problems for free by adding the Chrome extension — try it on your own worksheet before deciding if you need anything more.

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Photo Math Solver Solved

\text{Rectangle: } l = 12,\; w = 5. \text{ Find } A \text{ and } P.

1Area: $A = lw = 12 \cdot 5 = 60$ square units.
2Perimeter: $P = 2(12 + 5) = 2(17) = 34$ units.
3Note the different units — one is squared, the other is not.

On this page01Definition 02Formulas 03Methods 04Examples 05Practice 06Mistakes 07FAQ

01Definition

What are area and perimeter?

Perimeter is the distance all the way around a shape — add up every side. Area is the amount of flat surface inside it — how many unit squares would tile it. They answer different questions: perimeter is the fencing, area is the grass.

That difference shows up in the units and it is the fastest way to check your work. Perimeter is a single length, so it comes out in cm, m or in. Area multiplies two lengths together, so it comes out in cm², m² or in². If your area answer has no square unit on it, something is wrong.

Every plane-shape formula is a variation on "base times height". A rectangle is $bh$ exactly; a triangle is half of that because two copies make a rectangle; a trapezoid uses the average of its two parallel sides. Circles are the one genuine exception, where $\pi$ enters.

Where you'll actually use this

Flooring and paint: area of the room decides how much material to buy, perimeter decides the skirting board.
Fencing a garden is a perimeter problem; turfing it is an area problem — same plot, different formula.
Pizza value comparison: area grows with the square of the radius, so a 16-inch pizza has over twice the area of an 11-inch one.
Land and property listings quote area in m² or ft², computed by splitting irregular plots into rectangles.

Perimeter: The total distance around the outside edge of a plane shape.
Area: The amount of surface a shape covers, measured in square units.
Perpendicular height: The distance from the base to the opposite vertex measured at a right angle — never a slanted side.
Composite shape: A figure made from two or more standard shapes joined together or with a piece removed.
Circumference: The perimeter of a circle, $C = 2\pi r$ .

02Formulas

The formulas you need

Keep these on hand — every method below builds on them.

Rectangle

A = lw, \quad P = 2(l + w)

Square is the special case where l = w.

Triangle

A = \tfrac{1}{2}bh

h is the perpendicular height to the chosen base, not a slanted side.

Circle

A = \pi r^2, \quad C = 2\pi r

If given the diameter, halve it first: r = d/2.

Trapezoid

A = \tfrac{1}{2}(b_1 + b_2)h

Average of the two parallel sides, times the perpendicular height.

Parallelogram

A = bh

Same as a rectangle — the slant does not change the area.

Sector of a circle

A = \frac{\theta}{360°}\pi r^2

A fraction of the full circle, set by the angle at the centre.

03Methods

How to solve area and perimeter

Pick the method that fits the problem in front of you.

When to use it: A single standard shape with all the measurements labelled — the everyday case.

1Name the shape, then write down its formula before touching any numbers.
2Check the measurements match what the formula wants: a perpendicular height, not a slant; a radius, not a diameter.
3Substitute and evaluate, e.g. a triangle with $b = 10$ , $h = 6$ : $A = \frac{1}{2}(10)(6) = 30$ .
4Attach the unit — square units for area, plain units for perimeter.

04Worked examples

Area and Perimeter, solved step by step

From easy to hard — pick a problem to see its full solution.

\text{Rectangle: } l = 12,\; w = 5. \text{ Find } A \text{ and } P.

1Area: $A = lw = 12 \cdot 5 = 60$ square units.
2Perimeter: $P = 2(12 + 5) = 2(17) = 34$ units.
3Note the different units — one is squared, the other is not.

Try it on your own area and perimeter homework

Photo Math Solver reads an area or perimeter question straight from a screenshot — a labelled polygon, a circle with its radius marked, a composite floor plan — and shows which formula applies, the substitution with your numbers, and the answer in square or linear units. It handles the awkward cases too: composite shapes that need splitting, and reverse problems where the area is given and a side is missing.

05Practice

Now you try

Work each one on paper first, then check your answer.

Problem 1

\text{Parallelogram: } b = 9,\; h = 4

Problem 2

\text{Circle: } r = 5

Problem 3

\text{Square with perimeter } 36. \text{ Find its area.}

Problem 4

\text{Trapezoid: } b_1 = 6,\; b_2 = 10,\; h = 7

Problem 5

\text{A circle has area } 36\pi. \text{ Find its radius.}

Stuck on your area and perimeter homework? Screenshot it

Composite shapes are where area questions go wrong — an unlabelled side that has to be inferred by subtracting two others, or a dashed height that is easy to miss. Screenshot the whole figure with Photo Math Solver and the solution shows the split, the inferred lengths, and each piece’s area separately.

06Common mistakes

Where points get lost

Each of these shows up on real graded work — and each has a simple fix.

The mistake	Why it happens	The fix
Using a slanted side as the height in a triangle or parallelogram.	Diagrams label every side, and the slant is written right there while the perpendicular height is often a faint dashed line.	The height must meet the base at a right angle. Look for the dashed line and the small square — if only the slant is given, find the true height with the Pythagorean theorem first.
Squaring the diameter instead of the radius: computing $A = \pi(14)^2$ when $d = 14$ .	The number given in the problem gets substituted straight into the formula without checking which measurement it is.	Halve the diameter before anything else and write $r = 7$ down as its own step. This error inflates the area by a factor of four.
Forgetting the $\frac{1}{2}$ in the triangle formula.	The rectangle formula $bh$ is used far more often and takes over as the default.	Picture the triangle as half a rectangle with the same base and height. If your triangle area equals a rectangle’s, it is double what it should be.
Reporting area in plain units, or perimeter in square units.	The unit is added at the end from habit rather than from what the calculation actually did.	Count the multiplications: two lengths multiplied gives square units, sides added gives plain units. Many mark schemes deduct for this alone.
Including internal cut lines when finding the perimeter of a composite shape.	The same split that makes the area easy leaves a line through the middle of the figure that then gets measured.	Trace the outside boundary with your finger and add only the edges you actually travel along. The cut is a tool for area, not part of the outline.

Photo Math Solver shows every intermediate step, so slips like these are easy to catch before they cost you marks.

07FAQ