Continuing to Payment will take you to apayment page

Purchasing
SparkNotes PLUS
for a group?

Get Annual Plans at a discount when you buy 2 or more!

Price

$24.99$18.74/subscription + tax

Subtotal $37.48 + tax

Save 25%
on 2-49 accounts

Save 30%
on 50-99 accounts

Want 100 or more?
Contact us
for a customized plan.

Continuing to Payment will take you to apayment page

Your Plan

Payment Details

Payment Details

Payment Summary

SparkNotes Plus

You'll be billed after your free trial ends.

7-Day Free Trial

Not Applicable

Renews May 30, 2024May 23, 2024

Discounts (applied to next billing)

DUE NOW

US $0.00

SNPLUSROCKS20 | 20%Discount

This is not a valid promo code.

Discount Code(one code per order)

SparkNotes PLUS
Annual Plan - Group Discount

Qty: 00

SubtotalUS $0,000.00

Discount (00% off)
-US $000.00

TaxUS $XX.XX

DUE NOWUS $1,049.58

SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Choose Your Plan

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thank You!

You’ve successfully purchased a group discount. Your group members can use the joining link below to redeem their group membership. You'll also receive an email with the link.

No URL

Copy

Members will be prompted to log in or create an account to redeem their group membership.

Thanks for creating a SparkNotes account! Continue to start your free trial.

We're sorry, we could not create your account. SparkNotes PLUS is not available in your country. See what countries we’re in.

There was an error creating your account. Please check your payment details and try again.

Please wait while we process your payment

Your PLUS subscription has expired

We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.

Renew your subscription to regain access to all of our exclusive, ad-free study tools.

While there is no simple formula to determine area for most
quadrilaterals, and most
polygons for that matter, we saw last
section that for the special quadrilaterals
parallelograms and
trapezoids there are specific formulas for
determining area. The area of a triangle,
however, does. This is why it is so important that any polygon can be divided
into a number of triangles. The area of a polygon is equal to the sum of the
areas of all of the triangles within it.

The area of a triangle can be calculated in three ways. The most common
expression for the area of a triangle is one-half the product of the base
and the height (1/2AH). The height is formally called the altitude, and is
equal to the length of the line segment with
one endpoint at a vertex and the other endpoint on
the line that contains the
side opposite the vertex. Like all altitudes, this
segment must be perpendicular to the line
containing the side. The side opposite a given vertex is called the base of a
triangle. Here are some triangles pictured with their altitudes.

Another way to calculate the area of a triangle is called Heron's Formula, named
after the mathematician who first proved the formula worked. It is useful only
if you know the lengths of the sides of a triangle. The formula makes use of
the term semiperimeter. The semiperemeter of a triangle is equal to half
the sum of the lengths of the sides. Heron's Formula states that the area of a
triangle is equal to the square root of s(s-a)(s-b)(s-c), where s is the
semiperimeter of the triangle, and a, b, and c are the lengths of the three
sides. The proof of Heron's Formula is rather complex, and won't be discussed
here, but his formula works like a charm, especially if all that is known about
a triangle is the lengths of its sides.

The third and final way to calculate the area of a triangle has to do with the
angles as well as the side lengths of the
triangle. Any triangle has three sides and three angles. These are known as
the six parts of a triangle. Let the lengths of the sides of the triangle equal
a, b, and c. If the vertices opposite each length are angles of measure A, B,
and C, respectively, then the triangle would look like this:

The area of such a triangle is equal to one-half the product of the length of
two of the sides and the sine of their included angle. So in the triangle
above, Area=1/2(ab sin(C)), or Area=1/2(bc sin(A)), or Area=1/2(ac sin(B)).
The sine of an angle is a trigonometric property that you can learn more about
in the Trigonometry SparkNote.

With these formulas, it is possible to calculate the area of a triangle any time
you have any of the following information: 1) the length of the base and its
altitude; 2) the lengths of all three sides; or 3) the length of two sides and
the measure of their included angle. With these tools, it is possible to
calculate the area of triangles, and, as we shall see, by summing the areas of
triangles within a polygon, it is possible to calculate the area of any polygon.