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Area

Calculate the area of triangles.

Learning Objectives

After completing this section, you should be able to:

Calculate the area of triangles.
Calculate the area of quadrilaterals.
Calculate the area of other polygons.
Calculate the area of circles.

Area of Triangles

The formula for the area of a triangle is given as follows.

For example, consider the triangle in .

The base measures 4 cm and the height measures 5 cm. Using the formula, we can calculate the area: \[A=\frac{1}{2}(4)(5)=\frac{1}{2}(20)=10\ {\text{cm}}^{2}\]

In , the triangle has a base equal to 7 cm and a height equal to 3.5 cm. Notice that we can only find the height by dropping a perpendicular to the base. The area is then \[A=\frac{1}{2}(7)(3.5)=12.25\ {\text{cm}}^{2}.\]

Finding the Area of a Triangle

Try it.

Find the area of this triangle that has a base of 4 cm and the height is 6 cm ().

Solution

Using the formula, we have \[A=\frac{1}{2}(4)(6)=\frac{24}{2}=12\ {\text{cm}}^{2}\text{.}\]

Area of Quadrilaterals

A quadrilateral is a four-sided polygon with four vertices. Some quadrilaterals have either one or two sets of parallel sides. The set of quadrilaterals include the square, the rectangle, the parallelogram, the trapezoid, and the rhombus. The most common quadrilaterals are the square and the rectangle.

In , a $12\ \text{in}\times 12\ \text{in}$ grid is represented with twelve $1\ \text{in}\times 1\ \text{in}$ squares across each row, and twelve $1\ \text{in}\times 1\ \text{in}$ squares down each column. If you count the little squares, the sum equals 144 squares. Of course, you do not have to count little squares to find area—we have a formula. Thus, the formula for the area of a square, where $s=\text{length of a side}$, is $A=s⋅s$. The area of the square in is $A=12\ \text{in}\times 12\ \text{in}=144\ {\text{in}}^{2}.$

Rectangle

Similarly, the area for a rectangle is found by multiplying length times width. The rectangle in has width equal to 5 in and length equal to 12 in. The area is $A=5(12)=60\ {\text{in}}^{2}.$

Many everyday applications require the use of the perimeter and area formulas. Suppose you are remodeling your home and you want to replace all the flooring. You need to know how to calculate the area of the floor to purchase the correct amount of tile, or hardwood, or carpet. If you want to paint the rooms, you need to calculate the area of the walls and the ceiling to know how much paint to buy, and the list goes on. Can you think of other situations where you might need to calculate area?

Finding the Area of a Rectangle

Try it.

You have a garden with an area of 196 square feet. How wide can the garden be if the length is 28 feet?

Solution

The area of a rectangular region is $A=lw.$ Letting the width equal $w$: \[\begin{array}{lll}196 & = & 28w \\ \frac{196}{28} & = & w=7\ \text{ft}\end{array}\]

Determining the Cost of Floor Tile

Try it.

Jennifer is planning to install vinyl floor tile in her rectangular-shaped basement, which measures 29 ft by 16 ft. How many boxes of floor tile should she buy and how much will it cost if one box costs $50 and covers $20\ {\text{ft}}^{2}?$

Solution

The area of the basement floor is $A=29(16)=464\ {\text{ft}}^{2}.$ We will divide this area by $20\ \text{ft}.$ Thus, $\frac{464}{20}=23.2.$ Therefore, Jennifer will have to buy 24 boxes of tile at a cost $1,200.

Parallelogram

The area of a parallelogram can be found using the formula for the area of a triangle. Notice in , if we cut a diagonal across the parallelogram from one vertex to the opposite vertex, we have two triangles. If we multiply the area of a triangle by 2, we have the area of a parallelogram: \[\begin{array}{lll}A & = & 2(\frac{1}{2}bh) \\ A & = & bh\end{array}\]

For example, if we have a parallelogram with the base be equal to 10 inches and the height equal to 5 inches, the area will be $A=(10)(5)=50\ {\text{in}}^{2}.$

Finding the Area of a Parallelogram

Try it.

In the parallelogram (), if $FB=10,AD=15,$ find the exact area of the parallelogram.

Solution

Using the formula of $A=bh,$ we have \[A=10(15)=150\ {\text{cm}}^{2}.\]

Finding the Area of a Parallelogram Park

Try it.

The boundaries of a city park form a parallelogram (). The park takes up one city block, which is contained by two sets of parallel streets. Each street measures 55 yd long. The perpendicular distance between streets is 39 yd. How much sod, sold by the square foot, should the city purchase to cover the entire park and how much will it cost? The sod is sold for $0.50 per square foot, installation is $1.50 per square foot, and the cost of the equipment for the day is $100.

Solution

Step 1: As sod is sold by the square foot, the first thing we have to do is translate the measurements of the park from yards to feet. There are 3 ft to a yard, so 55 yd is equal to 165 ft, and 39 yd is equal to 117 ft.

Step 2: The park has the shape of a parallelogram, and the formula for the area is $A=bh$: \[A=165(117)=19,305\ {\text{ft}}^{2}.\]

Step 3: The city needs to purchase $19,305\ {\text{ft}}^{2}$ of sod. The cost will be $0.50 per square foot for the sod and $1.50 per square foot for installation, plus $100 for equipment: \[\begin{array}{lll}19,305(\text{\$}0.50)+19,305(\text{\$}1.50) & = & \text{\$}9,652.5+\text{\$}28,957.5+\text{\$}100.00 \\ & = & \text{\$}38,710.00\end{array}\]

Trapezoid

Another quadrilateral is the trapezoid. A trapezoid has one set of parallel sides or bases. The formula for the area of a trapezoid with parallel bases $a$ and $b$ and height $h$ is given here.

For example, find the area of the trapezoid in that has base $a$ equal to 8 cm, base $b$ equal to 6 cm, and height equal to 6 cm.

The area is $A=\frac{1}{2}(6)(6+8)=42\ {\text{cm}}^{2}$.

Finding the Area of a Trapezoid

Try it.

$ABCD$ () is a regular trapezoid with $\overset{\bar}{AB}‖\overset{\bar}{CD}.$ Find the exact perimeter of $ABCD$, and then find the area.

Solution

The perimeter is the measure of the boundary of the shape, so we just add up the lengths of the sides. We have $P=31+13.5+11+13.5=69\ \text{in}.$ Then, the area of the trapezoid using the formula is $A=\frac{1}{2}(10)(11+31)=210\ {\text{in}}^{2}$.

The rhombus has two sets of parallel sides. To find the area of a rhombus, there are two formulas we can use. One involves determining the measurement of the diagonals.

For our purposes here, we will use the formula that uses diagonals. For example, if the area of a rhombus is $220\ {\text{cm}}^{2},$ and the measure of ${d}_{2}=11,$ find the measure of ${d}_{1}.$ To solve this problem, we input the known values into the formula and solve for the unknown. See .

We have that \[\begin{array}{lll}220 & = & \frac{11{d}_{1}}{2} \\ 220(2) & = & 11{d}_{1} \\ \frac{440}{11} & = & {d}_{1}=40\end{array}\]

Finding the Area of a Rhombus

Try it.

Find the measurement of the diagonal ${d}_{1}$ if the area of the rhombus is $240\ {\text{cm}}^{2},$ and the measure of ${d}_{2}=24\ \text{cm}.$

Solution

Use the formula with the known values: \[\begin{array}{lll}240 & = & \frac{{d}_{1}(24)}{2} \\ 240(2) & = & {d}_{1}(24) \\ \frac{480}{24} & = & {d}_{1}=20\end{array}\]

Finding the Area of a Rhombus

Try it.

You notice a child flying a rhombus-shaped kite on the beach. When it falls to the ground, it falls on a beach towel measuring $36\ \text{in}$ by $72\ \text{in}.$ You notice that one of the diagonals of the kite is the same length as the $36\ \text{in}$ width of the towel. The second diagonal appears to be 2 in longer. What is the area of the kite ()?

Solution

Using the formula, we have: \[\begin{array}{lll}A & = & \frac{{d}_{1}{d}_{2}}{2} \\ & = & \frac{36(38)}{2}=684\ {\text{in}}^{2}\end{array}\]

Area of Polygons

To find the area of a regular polygon, we need to learn about a few more elements. First, the apothem $a$ of a regular polygon is a line segment that starts at the center and is perpendicular to a side. The radius $r$ of a regular polygon is also a line segment that starts at the center but extends to a vertex. See .

For example, consider the regular hexagon shown in with a side length of 4 cm, and the apothem measures $a=2\sqrt{3}.$

We have the perimeter, $p=6(4)=24\ \text{cm}.$ We have the apothem as $a=2\sqrt{3}.$ Then, the area is: \[\begin{array}{lll}A & = & \frac{1}{2}(2\sqrt{3})(24) \\ & = & 24\sqrt{3}=41.57\ {\text{cm}}^{2}\end{array}\]

Finding the Area of a Regular Octagon

Try it.

Find the area of a regular octagon with the apothem equal to 18 cm and a side length equal to 13 cm ().

Solution

Using the formula, we have the perimeter $p=8(13)=104\ \text{cm}.$ Then, the area is $A=\frac{1}{2}(10)(104)=936\ {\text{cm}}^{2}.$

Often, we have the need to change the units of one or more items in a problem to find a solution. For example, suppose you are purchasing new carpet for a room measured in feet, but carpeting is sold in terms of yards. You will have to convert feet to yards to purchase the correct amount of carpeting. Or, you may need to convert centimeters to inches, or feet to meters. In each case, it is essential to use the correct equivalency.

Changing Units

Try it.

Carpeting comes in units of square yards. Your living room measures 21 ft wide by 24 ft long. How much carpeting do you buy?

Solution

We must convert feet to yards. As there are 3 ft in 1 yd, we have $21\ \text{ft}=7\ \text{yds}$ and $24\ \text{ft}=8\ \text{yds}.$ Then, $7(8)=56\ {\text{yd}}^{2}.$

Area of Circles

Just as the circumference of a circle includes the number $\pi ,$ so does the formula for the area of a circle. Recall that $\pi$ is a non-terminating, non-repeating decimal number: $\pi =3.14159\ldots$. It represents the ratio of the circumference to the diameter, so it is a critical number in the calculation of circumference and area.

For example, to find the of the circle with radius equal to 3 cm, as shown in , is found using the formula $A=\pi {r}^{2}.$

We have \[\begin{array}{lll}A & = & \pi {r}^{2} \\ & = & \pi {(3)}^{2} \\ & = & 9\pi =28.27\ {\text{cm}}^{2}\end{array}\]

Finding the Area of a Circle

Try it.

Find the area of a circle with diameter of 16 cm.

Solution

The formula for the area of a circle is given in terms of the radius, so we cut the diameter in half. Then, the area is \[A=\pi {(8)}^{2}=201.1\ {\text{cm}}^{2}\text{.}\]

Determining the Better Value for Pizza

Try it.

You decide to order a pizza to share with your friend for dinner. The price for an 8-inch diameter pizza is $7.99. The price for 16-inch diameter pizza is $13.99. Which one do you think is the better value?

Solution

The area of the 8-inch diameter pizza is $A=\pi {(4)}^{2}=50.3\ {\text{in}}^{2}.$ The area of the 16-inch diameter pizza is $A=\pi {(8)}^{2}=201.1\ {\text{in}}^{2}.$ Next, we divide the cost of each pizza by its area in square inches. Thus, $\frac{13.99}{201.1}=\text{\$}0.07$ per square inch and $\frac{7.99}{50.3}=\text{\$}0.16$ per square inch. So clearly, the 16-inch pizza is the better value.

Applying Area to the Real World

Try it.

You want to purchase a tinted film, sold by the square foot, for the window in . (This problem should look familiar as we saw it earlier when calculating circumference.) The bottom part of the window is a rectangle, and the top part is a semicircle. Find the area and calculate the amount of film to purchase.

Solution

First, the rectangular portion has $A=5(10)=50\ {\text{ft}}^{2}$. For the top part, we have a semicircle with a diameter of 5 ft, so the radius is 2.5 ft. We want one half of the area of a circle with radius 2.5 ft, so the area of the top semicircle part is $A=\frac{1}{2}\pi {(2.5)}^{2}=9.8\ {\text{ft}}^{2}.$ Add the area of the rectangle to the area of the semicircle. Then, the total area to be covered with the window film is $A=50+9.8=59.8\ {\text{ft}}^{2}.$

Area within Area

Suppose you want to install a round hot tub on your backyard patio. How would you calculate the space needed for the hot tub? Or, let’s say that you want to purchase a new dining room table, but you are not sure if you have enough space for it. These are common issues people face every day. So, let’s take a look at how we solve these problems.

Finding the Area within an Area

Try it.

The patio in your backyard measures 20 ft by 10 ft (). On one-half of the patio, you have a 4-foot diameter table with six chairs taking up an area of approximately 36 sq feet. On the other half of the patio, you want to install a hot tub measuring 6 ft in diameter. How much room will the table with six chairs and the hot tub take up? How much area is left over?

Solution

The hot tub has a radius of 3 ft. That area is then $A=\pi {(3)}^{2}=9\pi =28.27\ {\text{ft}}^{2}.$ The total square feet taken up with the table and chairs and the hot tub is $36+28.27=64.27\ {\text{ft}}^{2}.$ The area left over is equal to the total area of the patio, $200\ {\text{ft}}^{2}$ minus the area for the table and chairs and the hot tub. Thus, the area left over is $200-64.27=135.7\ {\text{ft}}^{2}.$

Finding the Cost of Fertilizing an Area

Try it.

A sod farmer wants to fertilize a rectangular plot of land 150 ft by 240 ft. A bag of fertilizer covers $5,000\ {\text{ft}}^{2}$ and costs $200. How much will it cost to fertilize the entire plot of land?

Solution

The plot of land is $36,000\ {\text{ft}}^{2}.$ It will take 7.2 bags of fertilizer to cover the land area. Therefore, the farmer will have to purchase 8 bags of fertilizer at $200 a bag, which comes to $1,600.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

The area $A$ of a triangle is found with the formula $A=\frac{1}{2}bh,$ where $b$ is the base and $h$ is the height.
The area of a parallelogram is found using the formula $A=bh,$ where $b$ is the base and $h$ is the height.
The area of a rectangle is found using the formula $A=lw,$ where $l$ is the length and $w$ is the width.
The area of a trapezoid is found using the formula $A=\frac{1}{2}h({b}_{1}+{b}_{2}),$ where $h$ is the height, ${b}_{1}$ is the length of one base, and ${b}_{2}$ is the length of the other base.
The area of a rhombus is found using the formula $A=\frac{{d}_{1}{d}_{2}}{2},$ where ${d}_{1}$ is the length of one diagonal and ${d}_{2}$ is the length of the other diagonal.
The area of a regular polygon is found using the formula $A=\frac{1}{2}ap,$ where $a$ is the apothem and $p$ is the perimeter.
The area of a circle is found using the formula $A=\pi {r}^{2},$ where $r$ is the radius.

Formulas

The area of a triangle is given as $A=\frac{1}{2}bh,$ where $b$ represents the base and $h$ represents the height.

The formula for the area of a square is $A=s⋅s$ or $A={s}^{2}.$

The area of a rectangle is given as $A=lw.$

The area of a parallelogram is $A=bh.$

The formula for the area of a trapezoid is given as $A=\frac{1}{2}h(a+b).$

The area of a rhombus is found using one of these formulas:

$A=\frac{{d}_{1}{d}_{2}}{2},$ where ${d}_{1}$ and ${d}_{2}$ are the diagonals.
$A=\frac{1}{2}bh,$ where $b$ is the base and $h$ is the height.

The area of a regular polygon is found with the formula $A=\frac{1}{2}ap,$ where $a$ is the apothem and $p$ is the perimeter.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Practice (15)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Find the area of this triangle that has a base of 4 cm and the height is 6 cm ().

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

Using the formula, we have \[A=\frac{1}{2}(4)(6)=\frac{24}{2}=12\ {\text{cm}}^{2}\text{.}\]
You have a garden with an area of 196 square feet. How wide can the garden be if the length is 28 feet?

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

The area of a rectangular region is $A=lw.$ Letting the width equal $w$: \[\begin{array}{lll}196 & = & 28w \\ \frac{196}{28} & = & w=7\ \text{ft}\end{array}\]
Jennifer is planning to install vinyl floor tile in her rectangular-shaped basement, which measures 29 ft by 16 ft. How many boxes of floor tile should she buy and how much will it cost if one box costs $50 and covers $20\ {\text{ft}}^{2}?$

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

The area of the basement floor is $A=29(16)=464\ {\text{ft}}^{2}.$ We will divide this area by $20\ \text{ft}.$ Thus, $\frac{464}{20}=23.2.$ Therefore, Jennifer will have to buy 24 boxes of tile at a cost $1,200.
In the parallelogram (), if $FB=10,AD=15,$ find the exact area of the parallelogram.

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

Using the formula of $A=bh,$ we have \[A=10(15)=150\ {\text{cm}}^{2}.\]
The boundaries of a city park form a parallelogram (). The park takes up one city block, which is contained by two sets of parallel streets. Each street measures 55 yd long. The perpendicular distance between streets is 39 yd. How much sod, sold by the square foot, should the city purchase to cover the entire park and how much will it cost? The sod is sold for $0.50 per square foot, installation is $1.50 per square foot, and the cost of the equipment for the day is $100.

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

Step 1: As sod is sold by the square foot, the first thing we have to do is translate the measurements of the park from yards to feet. There are 3 ft to a yard, so 55 yd is equal to 165 ft, and 39 yd is equal to 117 ft.
Step 2: The park has the shape of a parallelogram, and the formula for the area is $A=bh$: \[A=165(117)=19,305\ {\text{ft}}^{2}.\]
Step 3: The city needs to purchase $19,305\ {\text{ft}}^{2}$ of sod. The cost will be $0.50 per square foot for the sod and $1.50 per square foot for installation, plus $100 for equipment: \[\begin{array}{lll}19,305(\text{\$}0.50)+19,305(\text{\$}1.50) & = & \text{\$}9,652.5+\text{\$}28,957.5+\text{\$}100.00 \\ & = & \text{\$}38,710.00\end{array}\]
$ABCD$ () is a regular trapezoid with $\overset{\bar}{AB}‖\overset{\bar}{CD}.$ Find the exact perimeter of $ABCD$, and then find the area.

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

The perimeter is the measure of the boundary of the shape, so we just add up the lengths of the sides. We have $P=31+13.5+11+13.5=69\ \text{in}.$ Then, the area of the trapezoid using the formula is $A=\frac{1}{2}(10)(11+31)=210\ {\text{in}}^{2}$.
Find the measurement of the diagonal ${d}_{1}$ if the area of the rhombus is $240\ {\text{cm}}^{2},$ and the measure of ${d}_{2}=24\ \text{cm}.$

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

Use the formula with the known values: \[\begin{array}{lll}240 & = & \frac{{d}_{1}(24)}{2} \\ 240(2) & = & {d}_{1}(24) \\ \frac{480}{24} & = & {d}_{1}=20\end{array}\]
You notice a child flying a rhombus-shaped kite on the beach. When it falls to the ground, it falls on a beach towel measuring $36\ \text{in}$ by $72\ \text{in}.$ You notice that one of the diagonals of the kite is the same length as the $36\ \text{in}$ width of the towel. The second diagonal appears to be 2 in longer. What is the area of the kite ()?

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

Using the formula, we have: \[\begin{array}{lll}A & = & \frac{{d}_{1}{d}_{2}}{2} \\ & = & \frac{36(38)}{2}=684\ {\text{in}}^{2}\end{array}\]
Find the area of a regular octagon with the apothem equal to 18 cm and a side length equal to 13 cm ().

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

Using the formula, we have the perimeter $p=8(13)=104\ \text{cm}.$ Then, the area is $A=\frac{1}{2}(10)(104)=936\ {\text{cm}}^{2}.$
Carpeting comes in units of square yards. Your living room measures 21 ft wide by 24 ft long. How much carpeting do you buy?

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

We must convert feet to yards. As there are 3 ft in 1 yd, we have $21\ \text{ft}=7\ \text{yds}$ and $24\ \text{ft}=8\ \text{yds}.$ Then, $7(8)=56\ {\text{yd}}^{2}.$
Find the area of a circle with diameter of 16 cm.

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

The formula for the area of a circle is given in terms of the radius, so we cut the diameter in half. Then, the area is \[A=\pi {(8)}^{2}=201.1\ {\text{cm}}^{2}\text{.}\]
You decide to order a pizza to share with your friend for dinner. The price for an 8-inch diameter pizza is $7.99. The price for 16-inch diameter pizza is $13.99. Which one do you think is the better value?

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

The area of the 8-inch diameter pizza is $A=\pi {(4)}^{2}=50.3\ {\text{in}}^{2}.$ The area of the 16-inch diameter pizza is $A=\pi {(8)}^{2}=201.1\ {\text{in}}^{2}.$ Next, we divide the cost of each pizza by its area in square inches. Thus, $\frac{13.99}{201.1}=\text{\$}0.07$ per square inch and $\frac{7.99}{50.3}=\text{\$}0.16$ per square inch. So clearly, the 16-inch pizza is the better value.
You want to purchase a tinted film, sold by the square foot, for the window in . (This problem should look familiar as we saw it earlier when calculating circumference.) The bottom part of the window is a rectangle, and the top part is a semicircle. Find the area and calculate the amount of film to purchase.

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

First, the rectangular portion has $A=5(10)=50\ {\text{ft}}^{2}$. For the top part, we have a semicircle with a diameter of 5 ft, so the radius is 2.5 ft. We want one half of the area of a circle with radius 2.5 ft, so the area of the top semicircle part is $A=\frac{1}{2}\pi {(2.5)}^{2}=9.8\ {\text{ft}}^{2}.$ Add the area of the rectangle to the area of the semicircle. Then, the total area to be covered with the window film is $A=50+9.8=59.8\ {\text{ft}}^{2}.$
The patio in your backyard measures 20 ft by 10 ft (). On one-half of the patio, you have a 4-foot diameter table with six chairs taking up an area of approximately 36 sq feet. On the other half of the patio, you want to install a hot tub measuring 6 ft in diameter. How much room will the table with six chairs and the hot tub take up? How much area is left over?

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

The hot tub has a radius of 3 ft. That area is then $A=\pi {(3)}^{2}=9\pi =28.27\ {\text{ft}}^{2}.$ The total square feet taken up with the table and chairs and the hot tub is $36+28.27=64.27\ {\text{ft}}^{2}.$ The area left over is equal to the total area of the patio, $200\ {\text{ft}}^{2}$ minus the area for the table and chairs and the hot tub. Thus, the area left over is $200-64.27=135.7\ {\text{ft}}^{2}.$
A sod farmer wants to fertilize a rectangular plot of land 150 ft by 240 ft. A bag of fertilizer covers $5,000\ {\text{ft}}^{2}$ and costs $200. How much will it cost to fertilize the entire plot of land?

ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

The plot of land is $36,000\ {\text{ft}}^{2}.$ It will take 7.2 bags of fertilizer to cover the land area. Therefore, the farmer will have to purchase 8 bags of fertilizer at $200 a bag, which comes to $1,600.

Symbols used here

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

\theta

theta
The usual name for an angle.

^\circ

degrees
1/360 of a full turn. 180° = π radians.

\sin,\ \cos,\ \tan

sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.

\angle ABC,\ \triangle ABC

angle, triangle
The angle at B between BA and BC; the triangle with those vertices.

\parallel,\ \perp,\ \cong,\ \sim

parallel, perpendicular, congruent, similar
Never meet; meet at 90°; identical shape and size; same shape.

How to: Area

Calculate the area of triangles.
Calculate the area of quadrilaterals.
Calculate the area of other polygons.
Calculate the area of circles.
triangle
square
rectangle
rhombus

Questions people ask

Why does every triangle have angles adding to 180°?

Draw a line through one vertex parallel to the opposite side: the two alternate angles equal the other two angles of the triangle, and the three angles at the vertex lie on a straight line. That is Euclid's proof, and it is why the fact holds only on a flat plane.

When do I use the law of sines versus the law of cosines?

Cosines when you know three sides, or two sides and the angle between them. Sines when you know an angle and the side opposite it, plus one more piece.

What is the difference between area and perimeter?

Perimeter is the length of the boundary (one dimension, measured in metres); area is the amount of surface inside (two dimensions, square metres). Doubling every side doubles the perimeter but quadruples the area.

ନିଜେ ଚେଷ୍ଟାକରନ୍ତୁ

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

ଅଧିକ Geometry

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