Math: Area, Volume, Circles and Trigonometry

This is Unit Fourteen of ACT Preparation: area, volume, circles and trigonometry. Unit thirteen worked with angles and the shapes of figures; this unit measures them. One question runs through it: is the answer a length, an area or a volume? The formula and the units follow from that. By the end of this part you will be able to find areas, including shaded regions. To use arcs, sectors and inscribed angles. To find volumes and surface areas. To read a circle's equation. And to use sine, cosine and tangent.

Five entries on area, from rectangles to shaded regions.

Perimeter and area.

ACT 1301 Rectangles and Triangles

Perimeter is the distance around a figure; area is the space it covers, in square units. A rectangle's area is length times width, and a triangle's is one half base times height: base ten and height six give thirty. The height is perpendicular to the base, not a slanted side.

Parallelograms and trapezoids.

ACT 1302 Parallelograms and Trapezoids

A parallelogram's area is base times height. A trapezoid's is one half the height times the sum of its two parallel bases: bases eight and twelve with height five give fifty. Use the perpendicular height for both, not the slanted side.

Figures built from pieces.

ACT 1303 Composite Figures

A composite figure's area is found by splitting it into simple shapes and adding, or by subtracting a cut-out from a larger shape. An L-shape of a ten by four and a four by three rectangle has area forty plus twelve, fifty-two. Count no piece twice.

The circle.

ACT 1304 Circumference and Area of a Circle

A circle's circumference is 2 pi r, or pi times the diameter, and its area is pi r squared. A diameter of ten gives a radius of five, a circumference of 10 pi, about 31.4, and an area of 25 pi, about 78.5. Area uses the radius, so halve a given diameter before squaring.

A shaded region, step by step.

ACT 1305 Area of a Shaded Region

Identify the outer shape and what is removed, compute each area, finding missing dimensions from the figure, then subtract, leaving pi in the answer unless a decimal is asked. A circle inside a square of side eight has diameter eight, radius four and area 16 pi, so the shaded corners are sixty-four minus 16 pi, about 13.7. Check which region is shaded: if the circle itself were shaded, the answer would be 16 pi.

Five entries on parts of a circle: tangents, arcs, sectors and inscribed angles.

A line that touches a circle.

ACT 1306 Tangent and Radius

A tangent touches a circle at one point and is perpendicular to the radius drawn there. From a point thirteen units from the center of a circle of radius five, the tangent has length twelve. The right angle at the point of tangency makes the Pythagorean theorem available.

Arcs and sectors.

ACT 1307 Arcs and Sectors

An arc's length and a sector's area are the fraction theta over 360 of the circle's circumference and area, where theta is the central angle. A ninety-degree angle in a circle of radius four marks an arc of 2 pi and a sector of 4 pi. Use the same fraction for both.

Arc length and sector area, step by step.

ACT 1308 Arc Length and Sector Area

Find the central angle and the radius, form the fraction of the circle, and multiply the circumference for an arc or the area for a sector; in radians, arc length is r theta. A pizza of radius nine cut into forty-degree slices gives one ninth: a crust of 2 pi, about 6.3 inches, and a slice of 9 pi, about 28.3 square inches. A question about the crust wants the arc; a question about the cheese wants the sector.

The inscribed angle.

ACT 1309 The Inscribed Angle

An inscribed angle has its vertex on the circle and equals half its intercepted arc, so half the central angle on the same arc. An arc of one hundred degrees gives an inscribed angle of fifty. An inscribed angle on a diameter intercepts 180 degrees, so it is always ninety.

Using an inscribed angle, step by step.

ACT 1310 Using an Inscribed Angle

Find the arc the angle intercepts, read its measure from a central angle on the same arc, then halve it, or double an inscribed angle to recover the arc. A central angle of 120 degrees makes the inscribed angle on the same arc sixty. In a triangle drawn with one side a diameter, the opposite angle is ninety, so angles of thirty-five and fifty-five complete it. Two inscribed angles on the same arc are equal.

Four entries on solids: what fills them and what covers them.

Boxes.

ACT 1311 Rectangular Solids

A rectangular solid's volume is length times width times height, in cubic units; its surface area is the total of its six faces. A five by three by two box has volume thirty and surface area sixty-two. Volume fills; surface area covers.

Cylinders.

ACT 1312 Cylinders

A cylinder's volume is the base area times the height, pi r squared h, and its surface area is two circles plus the wrapped side. A can of radius three and height ten holds 90 pi, about 283 cubic units. The side unrolls into a rectangle as wide as the circumference.

Spheres and cones.

ACT 1313 Spheres and Cones

A sphere's volume is four thirds pi r cubed and its surface area 4 pi r squared. A cone's volume is one third pi r squared h, a third of the matching cylinder. A sphere of radius three holds 36 pi. Cube the radius for volume; square it for surface area.

A solid, step by step.

ACT 1314 Volume and Surface Area of a Solid

Name the solid and decide whether the question asks what fills it or what covers it, read the dimensions, converting a diameter to a radius, and apply the formula, counting only exposed surfaces. A tank of diameter six and height eight holds 72 pi, about 226 cubic feet, and painting it outside, top and bottom, covers 66 pi, about 207 square feet. A tank open at the top has one circle, not two, giving 57 pi.

Five entries on circles on the coordinate plane, the conics, and right-triangle trigonometry.

The equation of a circle.

ACT 1315 The Circle Equation

A circle with center h, k and radius r has the equation x minus h, squared, plus y minus k, squared, equals r squared: the distance formula for every point on it. X minus two, squared, plus y plus one, squared, equals sixteen has center two, negative one and radius four. The right side is r squared.

Center and radius, step by step.

ACT 1316 Finding a Circle's Center and Radius

Group the x terms and the y terms with the constant on the right, complete the square for each, adding to both sides, and read the center and the radius. X squared plus y squared minus 6x plus 4y minus twelve equals zero becomes x minus three, squared, plus y plus two, squared, equals twenty-five: center three, negative two, radius five. Add each completing number to the right side too.

The other conic sections.

ACT 1317 Parabola, Ellipse and Hyperbola

In standard form, an ellipse adds x squared over a squared and y squared over b squared to make one; a hyperbola subtracts them; and a parabola has only one squared variable. The sign between the squares decides.

Sine, cosine and tangent.

ACT 1318 Sine, Cosine and Tangent

For an acute angle in a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent: soh-cah-toa. In a three-four-five triangle, the angle opposite three has sine three fifths, cosine four fifths and tangent three quarters. The ratios depend only on the angle, not the triangle's size.

Choosing the ratio, step by step.

ACT 1319 Choosing the Ratio

Mark the known angle, label the sides relative to it, and choose the ratio that links the side you know with the side you want. With a thirty-five-degree angle and an adjacent side of twenty, the opposite side uses tangent: twenty times the tangent of thirty-five, about fourteen. The hypotenuse uses cosine: twenty over the cosine of thirty-five, about 24.4. Opposite and adjacent change when the reference angle changes.

Look at what you can now do. You can find the area of any polygon or circle, and of a shaded region by subtracting. You can use tangents, arcs, sectors and inscribed angles. You can find what fills a solid and what covers it. You can read a circle's center and radius, completing the square when needed. And you can choose sine, cosine or tangent from the sides you know. Part two continues trigonometry: angles from sides, the unit circle, and any triangle.