Perimeter

Learn and practice calculating the perimeter of squares, rectangles, composite shapes, holes, and triangles.

Perimeter Study Guide

Learn and practice calculating the perimeter of squares, rectangles, composite shapes, holes, and triangles.

Practice Topics

1. Square & Rectangle

Learn and practice calculating the perimeter of squares, rectangles, composite shapes, and shapes with holes.

2. Right-angled Triangle

Learn and practice calculating the perimeter of right-angled triangles using the Pythagorean theorem.

3. Polygon

Learn and practice calculating the perimeter of polygons, composite polygons, and solving algebraic perimeter equations.

4. Circumference

Learn and practice calculating the circumference of circles, perimeter of sectors, rings, and racetracks.

5. Dimensions & Units

Learn and practice calculating perimeter with length unit conversions, and understand the difference between 1D and 2D scaling.

6. Caliper Measurement

Learn to measure shapes with a Vernier caliper, compute circumferences, and understand triangle SSS congruency and rectangle diagonal proofs.

Learning Guide

1. Square & Rectangle

The perimeter is the total length of the boundary of a two-dimensional shape. It represents the path that surrounds or outlines the shape.

1. The Full Boundary (Inner + Outer)

An important rule is that the perimeter is the full boundary of a shape. If a shape has a hole inside, walking along the entire boundary includes the outer boundary PLUS the inner boundary (the hole). Therefore, for shapes with holes:

2. Square and Rectangle Formulas

A rectangle has a width ww and a height hh. Opposite sides are equal, so the outer boundary has two sides of length ww and two sides of length hh. Summing them together gives the formula for the perimeter of a rectangle:
P=2w+2h=2(w+h)P = 2w + 2h = 2(w + h)

Variables:
• PP = Perimeter
• ww = width of the rectangle
• hh = height of the rectangle

Derivation:
1. Perimeter is the sum of all four side lengths: P=w+h+w+hP = w + h + w + h.
2. Group like terms: P=(w+w)+(h+h)=2w+2hP = (w + w) + (h + h) = 2w + 2h.
3. Factor out the common factor of 2: P=2(w+h)P = 2(w + h)
A square is a special case of a rectangle where all four sides are equal (w=h=sw = h = s). Substituting w=sw = s and h=sh = s into the rectangle formula, we derive:
P=2(s+s)=2(2s)=4sP = 2(s + s) = 2(2s) = 4s

Variables:
• PP = Perimeter
• ss = side length of the square

Derivation:
1. A square is a rectangle where width and height are equal: w=h=sw = h = s.
2. Substitute ss into the rectangle perimeter formula: P=2(s+s)=2(2s)=4sP = 2(s + s) = 2(2s) = 4s.

2. Right-angled Triangle

3. Right-angled Triangle Perimeter

For any triangle with sides aa, bb, and cc, the perimeter is simply the sum of all three sides:
P=a+b+cP = a + b + c

For a right-angled triangle with perpendicular legs aa and bb, we can use the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) to find the hypotenuse c=a2+b2c = \sqrt{a^2 + b^2}, and then calculate the perimeter.

Variables:
• PP = Perimeter
• aa = first perpendicular leg
• bb = second perpendicular leg
• cc = hypotenuse

Derivation:
1. The perimeter of any triangle is the sum of its three sides: P=a+b+cP = a + b + c.
2. For a right-angled triangle, if the hypotenuse cc is unknown, calculate it using the Pythagorean theorem: c=a2+b2c = \sqrt{a^2 + b^2}.
3. Substitute cc back to find the perimeter.

3. Polygon

4. Polygons and Sum of Sides

A polygon is any closed two-dimensional shape formed by straight line segments. The perimeter of a polygon is the total length of its boundary. To calculate the perimeter, we sum the lengths of all its sides:
s1s2s3s4s5
💡 Why do we use subscripts/indices (s1,s2,…s_1, s_2, \dots)?
In mathematics, when we have a group of similar items (like the sides of a polygon), we could label them with different letters like a,b,c,d,…a, b, c, d, \dots. But what if a polygon has 50 sides? The alphabet only has 26 letters!

To solve this, we use subscripts (indices). We choose one base letter (like ss for side) and attach a small index number (s1,s2,s3,…s_1, s_2, s_3, \dots):
• The base letter tells us what kind of item it is (all of them are sides).
• The index acts as a counter that identifies each specific side in our set.

This allows mathematicians to write clean, general formulas like P=s1+s2+⋯+snP = s_1 + s_2 + \dots + s_n that work for a polygon of any number of sides (nn) without running out of letters!

General Triangle as a Special Case

A triangle is a polygon with 3 sides. The triangle perimeter formula is just a special case of the general polygon perimeter formula when n=3n = 3:

Polygons with Holes

As we've already learned, if a polygon has holes inside, walking the entire boundary includes the outer boundary PLUS the inner boundaries of all the holes. The sides include all sides of both the outer and inner boundaries. For example, for a polygon with two holes, the perimeter is calculated by summing the outer boundary and the boundaries of both holes:
s1s2s3s4h1h2h3h4h5h6h712
P=Pouter+Phole 1+Phole 2P = P_{\text{outer}} + P_{\text{hole 1}} + P_{\text{hole 2}}
P=(s1+s2+s3+s4)+(h1+h2+h3)+(h4+h5+h6+h7)P = (s_1 + s_2 + s_3 + s_4) + (h_1 + h_2 + h_3) + (h_4 + h_5 + h_6 + h_7)

4. Circumference

5. Circumference and Sectors

The perimeter of a circle has a special name: circumference. It is the distance around the outside of the circle.
• The radius (rr) is the distance from the center to any point on the boundary.
• The diameter (dd) is the distance across the circle through the center (d=2rd = 2r).
rd

Perimeter of a Sector

A sector is a slice of a circle (like a pizza slice). The boundary of a sector consists of two straight radii and one curved arc.

By adding the two straight radius sides (2r2r) and the curved arc, the total perimeter formulas are:
rrθ

5. Dimensions & Units

1. What Is a Dimension?

A dimension tells us how many independent directions we need to measure a shape. Refer to the Dimensions & Units guide on the Area Page for details on dimensions and the metric/US customary systems.
• A line or boundary is 1-dimensional (1D) — it has only length.
• A flat surface is 2-dimensional (2D) — it has width and height.
• A solid object is 3-dimensional (3D).

2. Units of Perimeter

Because perimeter measures the boundary of a shape (a 1D length), it is always measured in linear units (e.g., mm, cmcm, ftft, inin), never in square units (which are for area). For example, if a rectangle has sides of 5 m5\text{ m} and 3 m3\text{ m}, its perimeter is 16 m16\text{ m} (linear), while its area is 15 m215\text{ m}^2 (square).
Rectangle (5m × 3m)5 m3 mArea = 15 m²(5 × 3 = 15)[2D Surface]Perimeter = 16 m(5 + 3 + 5 + 3 = 16) [1D Line]
Triangle (3m - 4m - 5m)3 m4 m5 mArea = 6 m²(½ × 4 × 3)[2D Surface]Perimeter = 12 m(3 + 4 + 5 = 12) [1D Line]
Circle (r = 3m)3 mArea ≈ 28.27 m²(π × 3²)[2D Surface]Perimeter ≈ 18.85 m(2 × π × 3) [1D Line]

3. Converting Length Units

To convert perimeter, use the standard length conversion formulas from the Formula Sheet:
• Metric: 1 m=100 cm=1,000 mm1\text{ m} = 100\text{ cm} = 1{,}000\text{ mm}, 1 cm=10 mm1\text{ cm} = 10\text{ mm}, 1 dm=10 cm1\text{ dm} = 10\text{ cm}.
• US Customary: 1 mi=1,760 yd1\text{ mi} = 1{,}760\text{ yd}, 1 yd=3 ft1\text{ yd} = 3\text{ ft}, 1 ft=12 in1\text{ ft} = 12\text{ in}.
• Multiply when converting from a larger unit to a smaller one (e.g., 3 ft×12=36 in3\text{ ft} \times 12 = 36\text{ in}). Divide when going from smaller to larger.

4. Perimeter vs. Area Scaling

When you enlarge a shape by a percentage (scaling all its dimensions by a scale factor kk):
• Perimeter (1D) scales linearly by kk. The ratio of the new perimeter to the old is k:1k:1.
• Area (2D) scales quadratically by k2k^2. The ratio of the new area to the old is k2:1k^2:1.

Example: If you increase all sides of a shape by 50%50\%, the scale factor is 1.5=3/21.5 = 3/2.
• The ratio of new perimeter to old perimeter is 3:2\mathbf{3:2}.
• The ratio of new area to old area is (3/2)2=9:4(3/2)^2 = \mathbf{9:4}.

5. Curiosity: The Rope Around the Earth

Imagine wrapping a rope tightly around the Earth's equator (assuming the Earth is a perfect sphere of radius RR). Its length is the circumference C1=2πRC_1 = 2\pi R.
Now, suppose you want to add just enough rope so that the rope hovers exactly 1 meter1\text{ meter} above the ground all the way around. The new radius is R+1 mR + 1\text{ m}, and the new circumference is C2=2π(R+1)=2πR+2πC_2 = 2\pi(R + 1) = 2\pi R + 2\pi.
The amount of rope you need to add is:
C2−C1=2π(R+1)−2πR=2π≈6.28 metersC_2 - C_1 = 2\pi(R + 1) - 2\pi R = 2\pi \approx 6.28\text{ meters}

Surprisingly, it does not depend on the radius of the Earth! Whether wrapping a tennis ball or the entire Earth, you only need to add about 6.28 m6.28\text{ m} of rope. This is because the relationship is linear (1-dimensional), a concept we will explore in future topics.

Mastering SealMath: Ratio & Unit Answers

• For ratio problems: enter your answer in a:b format (e.g., type 3:2 or 9:4). Make sure the ratio is fully simplified (no common factors).
• For calculation problems: enter your numerical answer as P = value. The unit is shown in the question, so do not type it.

6. Caliper Measurement

1. Measuring with a Caliper

A Vernier caliper has two scales: a fixed main scale and a sliding Vernier scale. Together, they measure dimensions with a precision of 0.1 mm0.1\text{ mm}.

How to read it:
1. Read the value on the main scale just to the left of the Vernier `0` mark.
2. Find the Vernier mark (0 to 10) that aligns perfectly with a main scale line.
3. Multiply that mark by 0.1 mm0.1\text{ mm} and add it to the main scale reading.
Vernier Caliper: Measuring 12.3 mm① Main Scale: 12 mm② Vernier: +0.3 mm (Aligns at 3)③ Total: 12.3 mm (Vernier 0)051012152025mmObject: 12.3 mm012345678910
12 mm+0.3 mm=12.3 mm
How it works (The Vernier Principle):
• Each division on the main scale is exactly 1 mm1\text{ mm}.
• The Vernier scale has 10 divisions spanning exactly 9 mm9\text{ mm} on the main scale, meaning each Vernier division is 0.9 mm0.9\text{ mm}.
• The difference between one main scale division (1 mm1\text{ mm}) and one Vernier division (0.9 mm0.9\text{ mm}) is exactly 0.1 mm0.1\text{ mm}.
• When the caliper opens by a fraction 0.y mm0.y\text{ mm}, the Vernier mark numbered yy aligns perfectly with a main-scale line.

• Example (Measuring 12.3 mm12.3\text{ mm}): The whole part is 12 mm12\text{ mm}. The fractional part is 0.3 mm0.3\text{ mm}. The 3rd Vernier mark is located 2.7 mm2.7\text{ mm} (3×0.9 mm3 \times 0.9\text{ mm}) to the right of the Vernier `0`. Since the caliper is open at 12.3 mm12.3\text{ mm}, this mark reaches 12.3+2.7=15.0 mm12.3 + 2.7 = 15.0\text{ mm}, aligning perfectly with the 15 mm15\text{ mm} line on the main scale.

2. Congruent Triangles (SSS Theorem)

Two triangles are congruent (≅\cong) if they have the exact same shape and size.
• SSS Theorem: If all three sides of one triangle are equal to the three sides of another, they are congruent.
• Ordering: The vertex order is crucial! △ABC≅△DEF\triangle ABC \cong \triangle DEF means vertex AA maps to DD, BB to EE, and CC to FF.
ABCcbaΔABC
DEFfedΔDEF
ΔABC≅ΔDEF\Delta ABC \cong \Delta DEF (SSS: a=d,b=e,c=fa=d, b=e, c=f)

3. Rectangle Diagonals & Angle Proof

Using geometric theorems, we can prove properties of rectangles:

• Diagonals are Equal: In a rectangle ABCDABCD (all angles 90∘90^\circ, opposite sides equal):
1. Triangles △ABC\triangle ABC and △BAD\triangle BAD are right-angled triangles that share leg ABAB, and have BC=ADBC=AD.
2. By the Pythagorean theorem, since their legs are equal, their hypotenuses (the diagonals) must be equal: AC=AB2+BC2=AB2+AD2=BDAC = \sqrt{AB^2 + BC^2} = \sqrt{AB^2 + AD^2} = BD.
3. Therefore, the diagonals are equal: AC=BDAC = BD.

• Converse Law (diagonals equal   ⟹  \implies rectangle): If opposite sides are equal (AB=CD,BC=DAAB=CD, BC=DA) and diagonals are equal (AC=BDAC=BD):
1. Triangles △ABC\triangle ABC and △CDA\triangle CDA share ACAC, and have AB=CD,BC=DAAB=CD, BC=DA. By SSS congruency, △ABC≅△CDA  ⟹  ∠B=∠D\triangle ABC \cong \triangle CDA \implies \angle B = \angle D.
2. Similarly, using diagonal BDBD, △ABD≅△CDB  ⟹  ∠A=∠C\triangle ABD \cong \triangle CDB \implies \angle A = \angle C.
3. Compare △ABC\triangle ABC and △BAD\triangle BAD. They share ABAB, have BC=ADBC=AD, and AC=BDAC=BD. By SSS congruency, △ABC≅△BAD  ⟹  ∠A=∠B\triangle ABC \cong \triangle BAD \implies \angle A = \angle B.
4. All four angles are equal: ∠A=∠B=∠C=∠D\angle A = \angle B = \angle C = \angle D. Since the angles sum to 360∘360^\circ, each must be 360∘/4=90∘360^\circ / 4 = 90^\circ.
Visual Proof: Diagonals are Equal ((AC=BD)(AC = BD))
ABCDRectangle ABCD
ABCcommon sideside hdiag ACTriangle ΔABC
ABDcommon sideside hdiag BDTriangle ΔBAD

Mastering SealMath: Special Symbols (∧, ≠)

• Logical AND (∧\land): Represents when multiple conditions must hold at the same time (e.g. AB=CD∧BC=DAAB=CD \land BC=DA). To type it, enter \land and press Enter, or use the shortcut land, or select ∧\land from the keyboard panel.

• Not Equal (≠\neq): Represents unequal segments or values (e.g. AB≠BCAB \neq BC). To type it, enter \neq and press Enter, or use the shortcut neq, or select ≠\neq from the keyboard panel (press Shift to find it).
Learning Topics

Frequently Asked Questions

What is perimeter?

Perimeter is the total boundary of a two-dimensional shape. It includes both the outer boundary and any inner boundaries (like the edges of holes inside the shape).

How do you calculate the perimeter of a rectangle?

The perimeter of a rectangle is the sum of all its four sides: P = 2w + 2h or P = 2(w + h), where w is the width and h is the height.

Why is a square's perimeter formula P = 4s?

A square is a special case of a rectangle where width and height are equal (w = h = s). Substituting this into the rectangle formula gives P = 2(s + s) = 4s.

How does a hole affect the perimeter?

Since perimeter measures the entire boundary of a shape, a hole adds to the perimeter. The total perimeter is the outer perimeter plus the inner perimeter (the perimeter of the hole).

How do you find the perimeter of a right-angled triangle if one side is missing?

Since we've already learned the Pythagorean theorem (a² + b² = c²), we can calculate the missing side length first and then sum all three sides together to get the perimeter.