How much paint covers a wall? How much wrapping paper does a gift box need? How much water fits in a fish tank? Each question is about a different measure: area covers a flat surface, surface area adds up every face of a solid, and volume counts how much space is inside. In this chapter you will use rectangles as your building blocks to measure triangles, parallelograms, trapezoids, composite shapes, prisms, and pyramids.
1. Area of triangles
Area is the number of square units that cover a flat shape. Rectangles are easy: \( A = \ell \times w \). A triangle is exactly half of a rectangle (or parallelogram) with the same base and height. Any side can serve as the base, and the height is the perpendicular distance from the base to the opposite vertex.
If a triangle has base \( b \) and height \( h \) (measured at a right angle to the base), then \[ A = \dfrac{1}{2}\, b\, h. \]
A flag is a triangle with a base of 14 ft and a height of 9 ft.
\( A = \dfrac{1}{2} \times 14 \times 9 = 7 \times 9 = 63 \). The flag has an area of 63 square feet (63 ft²).
The height must be perpendicular to the base. A slanted side of the triangle is not the height, so never multiply by it in the area formula.
2. Area of parallelograms and trapezoids
Cut a triangle off one end of a parallelogram and slide it to the other end: you get a rectangle with the same base and height. So a parallelogram has area \( A = b \times h \).
A trapezoid has one pair of parallel sides, called the bases \( b_1 \) and \( b_2 \). Two identical trapezoids fit together into a parallelogram whose base is \( b_1 + b_2 \), so one trapezoid is half of it.
| Shape | Area |
|---|---|
| Parallelogram | \( A = b \times h \) |
| Triangle | \( A = \dfrac{1}{2}\, b\, h \) |
| Trapezoid | \( A = \dfrac{b_1 + b_2}{2} \times h \) |
A trapezoid has bases of 7 cm and 13 cm and a height of 6 cm.
\( A = \dfrac{7 + 13}{2} \times 6 = 10 \times 6 = 60 \). The area is 60 cm².
3. Area of composite polygons
A composite polygon is built from simpler shapes. There are two good strategies: add the areas of pieces that fit together, or subtract the area of a missing piece from a bigger shape.
- Draw dashed lines to split the shape into rectangles, triangles, or trapezoids (or to complete it into one big rectangle).
- Find any missing lengths by adding or subtracting the sides you know.
- Compute each piece, then add (or subtract).
- Write the answer with square units.
The patio is a 10 by 7 rectangle with a 4 by 3 corner removed. Adding: Part 2 is \( 10 \times 4 = 40 \) and Part 1 is \( 6 \times 3 = 18 \), so \( A = 40 + 18 = 58 \). Subtracting: \( 10 \times 7 - 4 \times 3 = 70 - 12 = 58 \). Both methods give 58 square units, which is a great way to check your work.
4. Polygons on the coordinate plane
When the vertices of a polygon are ordered pairs \( (x, y) \), you can find side lengths by counting grid units. Two points with the same \( y \)-coordinate are on a horizontal line, and their distance is the difference of the \( x \)-coordinates. Two points with the same \( x \)-coordinate are on a vertical line, and the distance is the difference of the \( y \)-coordinates. Always take the larger number minus the smaller one; if one coordinate is negative, add the absolute values of the two numbers on opposite sides of zero.
Look at trapezoid ABCD with \( A(1, 1) \), \( B(7, 1) \), \( C(6, 4) \), and \( D(3, 4) \). Side AB is horizontal: \( 7 - 1 = 6 \). Side DC is horizontal: \( 6 - 3 = 3 \). The height is the vertical distance between the two parallel sides: \( 4 - 1 = 3 \).
\( A = \dfrac{6 + 3}{2} \times 3 = 4.5 \times 3 = 13.5 \). The area is 13.5 square units.
5. Nets of solids
A net is a flat pattern that folds up into a solid. The faces of a prism are two congruent bases and rectangular lateral faces. A pyramid has one base and triangular lateral faces that meet at a point called the apex.
Counting faces tells you the solid. A rectangular prism has 6 rectangles. A triangular prism has 2 triangles and 3 rectangles. A square pyramid has 1 square and 4 triangles.
On my home planet we say: “Unfold it, then count.” Sketch the net, label every face with its measurements, and nothing gets forgotten or counted twice!
6. Surface area of prisms and pyramids
The surface area of a solid is the sum of the areas of all its faces, which is the area of its net. It is measured in square units.
- Sketch the net and list every face.
- Find the area of each face.
- Add them all up (use symmetry: matching faces have equal areas).
- Give the answer in square units.
A box is 5 in long, 3 in wide, and 2 in tall. The faces come in three matching pairs: \( 5 \times 3 = 15 \), \( 5 \times 2 = 10 \), and \( 3 \times 2 = 6 \). So \( SA = 2(15 + 10 + 6) = 2 \times 31 = 62 \). The surface area is 62 in².
A pyramid has a square base of side 6 in and four triangular faces, each with base 6 in and slant height 5 in. The base is \( 6 \times 6 = 36 \). Each triangle is \( \dfrac{1}{2} \times 6 \times 5 = 15 \). So \( SA = 36 + 4 \times 15 = 96 \). The surface area is 96 in².
For the triangular faces of a pyramid, use the slant height (the height of the triangle), not the vertical height of the whole pyramid.
7. Volume of rectangular prisms with fractional edges
Volume measures the space inside a solid in cubic units. Fill a prism with unit cubes and count them: the volume of a rectangular prism is \( V = \ell \times w \times h \), or \( V = B \times h \) where \( B \) is the area of the base. This still works when the edges are fractions or mixed numbers.
A prism measures \( 2\dfrac{1}{2} \) ft by \( 1\dfrac{1}{2} \) ft by 2 ft. Pack it with cubes of edge \( \dfrac{1}{2} \) ft: there are \( 5 \times 3 \times 4 = 60 \) cubes. Each cube has volume \( \dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{8} \) ft³, so \( V = 60 \times \dfrac{1}{8} = 7.5 \) ft³. The formula agrees: \( \dfrac{5}{2} \times \dfrac{3}{2} \times 2 = \dfrac{30}{4} = 7.5 \). The volume is 7.5 cubic feet.
Length is in cm, area in cm², volume in cm³. Mixing them up is one of the most common mistakes, so check the unit before you write your answer.
Key takeaways
- Triangle: \( A = \dfrac{1}{2} b h \); parallelogram: \( A = b h \); trapezoid: \( A = \dfrac{b_1 + b_2}{2} h \). The height is always perpendicular to the base.
- For a composite polygon, split it into simple shapes and add, or subtract a missing piece.
- On the coordinate plane, subtract matching coordinates to get horizontal and vertical lengths.
- A net is a solid unfolded flat; count its faces to name the solid.
- Surface area = sum of the areas of all faces (square units). Use slant height for pyramid triangles.
- Volume of a rectangular prism: \( V = \ell \times w \times h \), even with fractional edges (cubic units).
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